Finite subgroups of the plane Cremona group

Igor V. Dolgachev^{1} and Vasily A. Iskovskikh^{2}

1 Department of Mathematics, University of Michigan, 525 E. University Av., Ann Arbor, Mi, 49109

  idolga@umich.edu

2 Steklov Institute of Mathematics, Gubkina 8, 117966, Moscow GSP-1, Russia

  iskovsk@mi.ras.ru

To Yuri I. Manin

Summary. This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane.

CONTENTS

1. Introduction

2. First examples

3. Rational G-surfaces

4. Automorphisms of minimal rational surfaces

5. Automorphisms of conic bundles

6. Automorphisms of Del Pezzo surfaces

7. Elementary links and the factorization theorem

8. Birational classes of minimal G-surfaces

9. What is left?

10. Tables

1 Introduction

The Cremona group Cr_{k}(n) over a field k is the group of birational automorphisms of the projective space ℙ_{k}^{n}, or, equivalently, the group of kautomorphisms of the field k(x_{1}, x_{2}, …, x_{n}) of rational functions in n independent variables. The group Cr_{k}(1) is the group of automorphisms of the projective line, and hence it is isomorphic to the projective linear group PGL_{k}(2). Already in the case n = 2 the group Cr_{k}(2) is not well understood in spite of extensive classical literature (e.g. [20], [34]) on the subject as well as some modern research and expositions of classical results (e.g. [2]). Very little is known about the Cremona groups in higher-dimensional spaces.

In this paper we consider the plane Cremona group over the field of complex numbers, denoted by Cr(2). We return to the classical problem of classification of finite subgroups of Cr(2). The classification of finite subgroups of PGL(2) is well-known and goes back to F. Klein. It consists of cyclic, dihedral, tetrahedral, octahedral and icosahedral groups. Groups of the same type and order constitute a unique conjugacy class in PGL(2). Our goal is to find a similar classification in the two-dimensional case.

The history of this problem begins with the work of E. Bertini [9] who classified conjugacy classes of subgroups of order 2 in Cr(2). Already in this case the answer is drastically different. The set of conjugacy classes is parametrized by a disconnected algebraic variety whose connected components are respectively isomorphic to either the moduli spaces of hyperelliptic curves of genus g (de Jonquiéres involutions), or the moduli space of canonical curves of genus 3 (Geiser involutions), or the moduli space of canonical curves of genus 4 with vanishing theta characteristic (Bertini involutions). Bertini’s proof was considered to be incomplete even according to the standards of rigor of the 19th century algebraic geometry. A complete and short proof was published only a few years ago by L. Bayle and A. Beauville [5].

In 1894 G. Castelnuovo [15], as an application of his theory of adjoint linear systems, proved that any element of finite order in Cr(2) leaves invariant either a net of lines, or a pencil of lines, or a linear system of cubic curves with n ≤ 8 base points. A similar result was claimed earlier by S. Kantor in his memoir which was awarded a prize by the Accademia delle Scienze di Napoli in 1883. However Kantor’s arguments, as was pointed out by Castelnuovo, required justifications. Kantor went much further and announced a similar theorem for arbitrary finite subgroups of Cr(2). He proceeded to classify possible groups in each case (projective linear groups, groups of de Jonquiéres type, and groups of type M_{n}). A much clearer exposition of his results can be found in a paper of A. Wiman [49]. Unfortunately, Kantor’s classification, even with some correction made by Wiman, is incomplete for the following two reasons. First, only maximal groups were considered and even some of them were missed. The most notorious example is a cyclic group of order 8 of automorphisms of a cubic surface, also missed by B. Segre [47] (see [33]). Second, although Kantor was aware of the problem of conjugacy of subgroups, he did not attempt to fully investigate this problem.

The goal of our work is to complete Kantor’s classification. We use a modern approach to the problem initiated in the works of Yuri Manin and the second author (see a survey of their results in [38]). Their work gives a clear understanding of the conjugacy problem via the concept of a rational G-surface. It is a pair (S, G) consisting of a nonsingular rational projective surface and a subgroup G of its automorphism group. A birational map S− → ℙ_{k}^{2} realizes G as a finite subgroup of Cr(2). Two birational isomorphic Gsurfaces define conjugate subgroups of Cr(2), and conversely a conjugacy class of a finite subgroup G of Cr(2) can be realized as a birational isomorphism class of G-surfaces. In this way classification of conjugacy classes of subgroups of Cr(2) becomes equivalent to the birational classification of G-surfaces. A G-equivariant analog of a minimal surface allows one to concentrate on the study of minimal G-surfaces, i.e. surfaces which cannot be G-equivariantly birationally and regularly mapped to another G-surface. Minimal G-surfaces turn out to be G-isomorphic either to the projective plane, or a conic bundle, or a Del Pezzo surface of degree d = 9 − n ≤ 6. This leads to groups of projective transformations, or groups of de Jonquiéres type, or groups of type M_{n}, respectively. To complete the classification one requires

• to classify all finite groups G which may occur in a minimal G-pair (S, G);

• to determine when two minimal G-surfaces are birationally isomorphic.

To solve the first part of the problem one has to compute the full automorphism group of a conic bundle surface or a Del Pezzo surface (in the latter case this was essentially accomplished by Kantor and Wiman), then make a list of all finite subgroups which act minimally on the surface (this did not come up in the works of Kantor and Wiman). The second part is less straightforward. For this we use the ideas from Mori’s theory to decompose a birational map of rational G-surfaces into elementary links. This theory was successfully applied in the arithmetic case, where the analog of the group G is the Galois group of the base field (see [38]). We borrow these results with obvious modifications adjusted to the geometric case. Here we use the analogy between k-rational points in the arithmetic case (fixed points of the Galois action) and fixed points of the G-action. As an important implication of the classification of elementary G-links is the rigidity property of groups of type M_{n} with n ≥ 6: any minimal Del Pezzo surface (S, G) of degree d ≤ 3 is not isomorphic to a minimal G-surface of different type. This allows us to avoid much of the painful analysis of possible conjugacy for a lot of groups.

The large amount of group-theoretical computations needed for the classification of finite subgroups of groups of automorphisms of conic bundles and Del Pezzo surfaces makes us expect some possible gaps in our classification. This seems to be a destiny of enormous classification problems. We hope that our hard work will be useful for the future faultless classification of conjugacy classes of finite subgroups of Cr(2).

It is appropriate to mention some recent work on classification of conjugacy classes of subgroups of Cr(2). We have already mentioned the work of L. Bayle and A. Beauville on groups of order 2. The papers [7],[22], [51] study groups of prime orders, Beauville’s paper [8] classifies p-elementary groups, and a thesis of J. Blanc [6] contains a classification of all finite abelian groups. The second author studies two non-conjugate classes of subgroups isomorphic to S_{3} × ℤ/2ℤ. In the work of S. Bannai and H. Tokunaga examples are given of non-conjugate subgroups isomorphic to S_{4} and A_{5}.

This paper is partly based on the lectures by the first author in workshops on Cremona transformations in Torino in September 2005 and Lisbon in May 2006. He takes the opportunity to thank the organizers for the invitation and for providing a good audience. We both like to thank A. Beauville, Chenyang Xu and, especially, J. Blanc for pointing out some errors in previous versions of our paper.

The first author was supported in part by NSF grant 0245203. The second author was supported in part by RFBR 05-01-00353-a RFBR 06-01-72017-MNTI-a, grant CRDF RUMI 2692-MO-05 and grant NSh 9969 2006 1.

2 First examples

2.1 Homaloidal linear systems

We will be working over the field of complex numbers. Recall that a dominant rational map χ : ℙ^{2}− → ℙ^{2} is given by a 2-dimensional linear system H equal to the proper transform of the linear system of lines H′ = |ℓ| in the target plane. A choice of a basis in H gives an explicit formula for the map in terms of homogeneous coordinates

(x′_{0}, x′_{1}, x′_{2}) = (P_{0}(x_{0}, x_{1}, x_{2}), P_{1}(x_{0}, x_{1}, x_{2}), P_{2}(x_{0}, x_{1}, x_{2})),

where P_{0}, P_{1}, P_{2} are linear independent homogeneous polynomials of degree d, called the (algebraic) degree of the map. This is the smallest number d such that H is contained in the complete linear system |O_{ℙ^{2}}(d)| of curves of degree d in the plane. By definition of the proper transform, the linear system H has no fixed components, or, equivalently, the polynomials P_{i}’s are mutually coprime. The birational map χ is not a projective transformation if and only if the degree is larger than 1, or, equivalently, when χ has base points, the common zeros of the members of the linear system. A linear system defining a birational map is called a homaloidal linear system. Being proper transform of a general line under a birational map, its general member is an irreducible rational curve. Also two general curves from the linear system intersect outside the base points at one point. These two conditions characterize homaloidal linear systems (more about this later).

2.2 Quadratic transformations

A quadratic Cremona transformation is a birational map χ : ℙ^{2}− → ℙ^{2} of degree 2. The simplest example is the standard quadratic transformation defined by the formula

τ_{1} : (x_{0}, x_{1}, x_{2}) ↦ (x_{1}x_{2}, x_{0}x_{2}, x_{0}x_{1}).  (1)

In affine coordinates this is given by τ_{1} : (x, y) ↦ (\frac{1}{x}, \frac{1}{y}). It follows from the definition that τ_{1}^{−1} = τ_{1}, i.e., τ_{1} is a birational involution of ℙ^{2}. The base points of T are the points p_{0} = (1, 0, 0), p_{1} = (0, 1, 0), p_{2} = (0, 0, 1). The transformation maps an open subset of the coordinate line x_{i} = 0 to the point p_{i}. The homaloidal linear system defining τ_{1} is the linear system of conics passing through the points p_{0}, p_{1}, p_{2}.

The Moebius transformation x ↦ x^{−1} of ℙ^{1} is conjugate to the transformation x ↦ −x (by means of the map x ↦ \frac{x−1}{x+1}). This shows that the standard Cremona transformation τ_{1} is conjugate in Cr(2) to a projective transformation given by

(x_{0}, x_{1}, x_{2}) ↦ (x_{0}, −x_{1}, −x_{2}).

When we change the homaloidal linear system defining τ_{1} to the homaloidal linear system of conics passing through the point p_{1}, p_{2} and tangent at p_{2} to the line x_{0} = 0 we obtain the transformation

τ_{2} : (x_{0}, x_{1}, x_{2}) ↦ (x_{1}^{2}, x_{0}x_{1}, x_{0}x_{2}).  (2)

In affine coordinates it is given by (x, y) ↦ (\frac{1}{x}, \frac{y}{x^{2}} ). The transformation τ_{2} is also a birational involution conjugate to a projective involution. To see this we define a rational map χ : ℙ^{2}− → ℙ^{3} by the formula (x_{0}, x_{1}, x_{2}) ↦ (x_{1}^{2}, x_{0}x_{1}, x_{0}x_{2}, x_{1}x_{2}). The Cremona transformation τ_{2} acts on ℙ^{3} via this transformation by (u_{0}, u_{1}, u_{2}, u_{3}) ↦ (u_{1}, u_{0}, u_{3}, u_{2}). Composing with the projection of the image from the fixed point (1, 1, 1, 1) we get a birational map (x_{0}, x_{1}, x_{2}) ↦ (y_{0}, y_{1}, y_{2}) = (x_{1}(x_{0} − x_{1}), x_{0}x_{2} − x_{1}^{2}, x_{1}(x_{2} − x_{1})). It defines the conjugation of τ_{2} with the projective transformation (y_{0}, y_{1}, y_{2}) ↦ (−y_{0}, y_{2} − y_{0}, y_{1} − y_{0}).

Finally, we could further “degenerate” τ_{1} by considering the linear system of conics passing through the point p_{3} and intersecting each other at this point with multiplicity 3. This linear system defines a birational involution

τ_{3} : (x_{0}, x_{1}, x_{2}) ↦ (x_{0}^{2}, x_{0}x_{1}, x_{1}^{2} − x_{0}x_{2}).  (3)

Again it can be shown that τ_{3} is conjugate to a projective involution.

Recall that a birational transformation is not determined by the choice of a homaloidal linear system, one has to choose additionally a basis of the linear system. In the above examples, the basis is chosen to make the transformation an involution.

2.3 De Jonquiéres involutions

Here we exhibit a series of birational involutions which are not conjugate to each other and not conjugate to a projective involution. In affine coordinates they are given by the formula

dj_{P} : (x, y) ↦ (x, \frac{P (x)}{y}),  (4)

where P (x) a polynomial of degree 2g + 1 or 2g + 2 without multiple roots. The conjugation by the transformation (x, y) ↦ (\frac{ax+b}{cx+d}, y) shows that the conjugacy class of dj_{P} depends only on the orbit of the set of roots of P with respect to the group PGL(2), or, in other words, on the birational class of the hyperelliptic curve

y^{2} + P (x) = 0.  (5)

The transformation dj_{P} has the following beautiful geometric interpretation. Consider the projective model H_{g+2} of the hyperelliptic curve (5) given by the homogeneous equation of degree g + 2

T_{2}^{2}F_{g}(T_{0}, T_{1}) + 2T_{2}F_{g+1}(T_{0}, T_{1}) + F_{g+2}(T_{0}, T_{1}) = 0,  (6)

where

D = F_{g+1}^{2} − F_{g}F_{g+2} = T_{0}^{2g+2}P (T_{1}/T_{0})

is the homogenization of the polynomial P (x). The curve H_{g+2} has an ordinary singular point of multiplicity g at q = (0, 0, 1) and the projection from this point to ℙ^{1} exhibits the curve as a double cover of ℙ^{1} branched over the 2g +2 zeroes of D.

Consider the affine set T_{2} = 1 with affine coordinates (x, y) = (T_{0}/T_{2}, T_{1}/T_{2}). A general line y = kx intersects the curve H_{g+2} at the point q = (0, 0) with multiplicity g and at two other points (α, kα) and (α′, kα′), where α, α′ are the roots of the quadratic equation

t^{2}F_{g+2}(1, k) + 2tF_{g+1}(1, k) + F_{g}(1, k) = 0.

Take a general point p = (x, kx) on the line and define the point p′ = (x′, kx′) such that the pairs (α, kα), (α′, kα′) and (x, kx), (x′, kx′) are harmonic conjugate. This means that x, x′ are the roots of the equation at^{2}+2bt+c = 0, where aF_{g}(1, k) + cF_{g+2}(1, k) − 2bF_{g+1}(1, k) = 0. Since x + x′ = −2b/c, xx′ = c/a we get F_{g}(1, k) + xx′F_{g+2}(1, k) + (x + x′)F_{g+1}(1, k) = 0. We express x′ as (ax + b)/(cx + d) and solve for (a, b, c, d) to obtain

x′ = \frac{−F_{g+1}(1, k)x − F_{g}(1, k)}{xF_{g+2}(1, k) + F_{g+1}(1, k)}.

Since k = y/x, after changing the affine coordinates (x, y) = (x_{0}/x_{2}, x_{1}/x_{2}) to (X, Y) = (x_{1}/x_{0}, x_{2}/x_{0}) = (y/x, 1/x), we get

IH_{g+2} : (X, Y) ↦ (X′, Y′) := (X, \frac{−Y P_{g+1}(X) − P_{g+2}(X)}{P_{g}(X)Y + P_{g+1}(X)}),  (7)

where P_{i}(X) = F_{i}(1, X). Let T : (x, y) ↦ (x, yP_{g} + P_{g+1}). Taking P (x) = P_{g+1}^{2} − P_{g}P_{g+2}, we check that T^{−1} ∘ dj_{P} ∘ T = IH_{g+2}. This shows that our geometric de Jonquiéres involution IH_{g+2} given by (7) is conjugate to the de Jonquiéres involution dj_{P} defined by (4).

Let us rewrite (7) in homogeneous coordinates:

x′_{0} = x_{0}(x_{2}F_{g}(x_{0}, x_{1}) + F_{g+1}(x_{0}, x_{1})),  (8)
x′_{1} = x_{1}(x_{2}F_{g}(x_{0}, x_{1}) + F_{g+1}(x_{0}, x_{1})),
x′_{2} = −x_{2}F_{g+1}(x_{0}, x_{1}) − F_{g+2}(x_{0}, x_{1}).

Now it is clear that the homaloidal linear system defining IH_{g+2} consists of curves of degree g + 2 which pass through the singular point q of the hyperelliptic curve (6) with multiplicity g. Other base points satisfy

x_{2}F_{g}(x_{0}, x_{1}) + F_{g+1}(x_{0}, x_{1}) = −x_{2}F_{g+1}(x_{0}, x_{1}) − F_{g+2}(x_{0}, x_{1}) = 0.

Eliminating x_{2}, we get the equation F_{g+1}^{2}−F_{g}F_{g+2} = 0 which defines the set of the 2g +2 ramification points p_{1}, …, p_{2g+2} of the projection H_{g+2} \{q} → ℙ^{1}.

Let

Γ : T_{2}F_{g}(T_{0}, T_{1}) + F_{g+1}(T_{0}, T_{1}) = 0

be the first polar Γ of H_{g+2} with respect to the point q. The transformation IH_{g+2} blows down Γ and the lines ⟨q, p_{i}⟩ to points. It follows immediately from (8) that the set of fixed points of the involution IH_{g+2} outside the base locus is equal to the hyperelliptic curve (6). Also we see that the pencil of lines through q is invariant with respect to IH_{g+2}.

Let σ : S → ℙ^{2} be the blow-up of the point q and the points p_{1}, …, p_{2g+2}. The full pre-image of the line ℓ_{i} = ⟨q, p_{i}⟩ consists of two irreducible components, each isomorphic to ℙ^{1}. They intersect transversally at one point. We will call such a reducible curve, a bouquet of two ℙ^{1}’s. One component is the exceptional curve R_{i} = σ^{−1}(p_{i}) and another one is the proper transform R′_{i} of the line ℓ_{i}. The proper transform of H_{g+2} intersects σ^{−1}(ℓ_{i}) at its singular point. Thus the proper transform H̄_{g+2} of the hyperelliptic curve H_{g+2} intersects the exceptional curve E = σ^{−1}(q) at the same points where the proper transform of lines ℓ_{i} intersect E. The proper transform Γ̄ of Γ intersects R_{i} at one nonsingular point, and intersects E at g points, the same points where the proper inverse transform H̄_{g+2} of H_{g+2} intersects E. The involution IH_{g+2} lifts to a biregular automorphism τ of S. It switches the components R_{i} and R′_{i} of σ^{−1}(ℓ_{i}), switches E with Γ̄ and fixes the curve H̄_{g+2} pointwiseley. The pencil of lines through q defines a morphism ϕ : S → ℙ^{1} whose fibres over the points corresponding to the lines ℓ_{i} are isomorphic to a bouquet of two ℙ^{1}’s. All other fibres are isomorphic to ℙ^{1}. This is an example of a conic bundle or a Mori fibration (or in the archaic terminology of [37], a minimal rational surface with a pencil of rational curves).

To show that the birational involutions IH_{g+2}, g > 0, are not conjugate to each other or to a projective involution we use the following.

Lemma 2.1. Let G be a finite subgroup of Cr(2) and let C_{1}, …, C_{k} be nonrational irreducible curves on ℙ^{2} such that each of them contains an open subset C_{i}^{0} whose points are fixed under all g ∈ G. Then the set of birational isomorphism classes of the curves C_{i} is an invariant of the conjugacy class of G in Cr(2).

[figure: Γ; p_{1}; p_{2}; q; E; H̄_{2}; Γ̄; R_{1}; R′_{1}; R_{2}; R′_{2}; ⟵; σ; H_{2}]

Proof. Suppose G = T ∘ G′ ∘ T^{−1} for some subgroup G′ of Cr(2) and some T ∈ Cr(2). Then, replacing C_{i}^{0} by a smaller open subset we may assume that T^{−1}(C_{i}^{0}) is defined and consists of fixed points of G′. Since C_{i} is not rational, T^{−1}(C_{i}^{0}) is not a point, and hence its Zariski closure is a rational irreducible curve C′_{i} birationally isomorphic to C_{i} which contains an open subset of fixed points of G′.

Since a connected component of the fixed locus of a finite group of projective transformations is a line or a point, we see that IH_{g+2} is not conjugate to a subgroup of projective transformations for any g > 0. Since IH_{g+2} is conjugate to some involution (4), where P (x) is determined by the birational isomorphism class of H_{g+2}, we see from the previous lemma that IH_{g+2} is conjugate to IH′_{g′+2} if and only if g = g′ and the curves H_{g+2} and H′_{g+2} are birationally isomorphic. Finally, let us look at the involution IH_{2}. It is a quadratic transformation which is conjugate to the quadratic transformation τ_{2} : (x, y) ↦ (x, x/y).

A Jonquiéres involution (4) is a special case of a Cremona transformation of the form

(x, y) ↦ (\frac{ax + b}{cx + d}, \frac{r_{1}(x)y + r_{2}(x)}{r_{3}(x)y + r_{4}(x)}),

where a, b, c, d ∈ ℂ, ad − bc ≠ 0 and r_{i}(x) ∈ ℂ(x) with r_{1}(x)r_{4}(x) − r_{2}(x)r_{3}(x) ≠ 0. These transformations form a subgroup of Cr(2) called a de Jonquiéres subgroup and denoted by dJ(2). Of course, its definition requires a choice of a transcendence basis of the field ℂ(ℙ^{2}). If we identify Cr(2) with the group Aut_{ℂ}(ℂ(x, y)), and consider the field ℂ(x, y) as a field K(y), where K = ℂ(x), then

dJ(2) ≅ PGL_{ℂ(x)}(2) ⋊ PGL(2)

where PGL(2) acts on PGL_{ℂ(x)}(2) via Moebius transformations of the variable x.

It is clear that all elements from dJ(2) leave the pencil of lines parallel to the y-axis invariant. One can show that a subgroup of Cr(2) which leaves a pencil of rational curves invariant is conjugate to dJ(2).

2.4 Geiser and Bertini involutions

The classical definition of a Geiser involution is as follows [29]. Fix 7 points p_{1}, …, p_{7} in ℙ^{2} in general position (we will make this more precise later). The linear system L of cubic curves through the seven points is two-dimensional. Take a general point p and consider the pencil of curves from L passing through p. Since a general pencil of cubic curves has 9 base points, we can define γ(p) as the ninth base point of the pencil. One can show that the algebraic degree of a Geiser involution is equal to 8. Another way to see a Geiser involution is as follows. The linear system L defines a rational map of degree 2

f : ℙ^{2}− → |L|^{∗} ≅ ℙ^{2}.

The points p and γ(p) lie in the same fibre. Thus γ is a birational deck transformation of this cover. Blowing up the seven points, we obtain a Del Pezzo surface S of degree 2 (more about this later), and a regular map of degree 2 from S to ℙ^{2}. The Geiser involution γ becomes an automorphism of the surface S.

It is easy to see that the fixed points of a Geiser involution lie on the ramification curve of f. This curve is a curve of degree 6 with double points at the points p_{1}, …, p_{7}. It is birationally isomorphic to a canonical curve of genus 3. Applying Lemma 2.1, we obtain that a Geiser involution is not conjugate to any de Jonquiéres involution IH_{g+2}. Also, as we will see later, the conjugacy classes of Geiser involutions are in a bijective correspondence with the moduli space of canonical curves of genus 3.

To define a Bertini involution we fix 8 points in ℙ^{2} in general position and consider the pencil of cubic curves through these points. It has the ninth base point p_{9}. For any general point p there will be a unique cubic curve C(p) from the pencil which passes through p. Take p_{9} for the zero in the group law of the cubic C(p) and define β(p) as the negative −p with respect to the group law. This defines a birational involution on ℙ^{2}, a Bertini involution [9]. One can show that the algebraic degree of a Bertini involution is equal to 17. We will see later that the fixed points of a Bertini involution lie on a canonical curve of genus 4 with vanishing theta characteristic (isomorphic to a nonsingular intersection of a cubic surface and a quadric cone in ℙ^{3}). So, a Bertini involution is not conjugate to a Geiser involution or a de Jonquiéres involution. It can be realized as an automorphism of the blow-up of the eight points (a Del Pezzo surface of degree 1), and the quotient by this involution is isomorphic to a quadratic cone in ℙ^{3}.

3 Rational G-surfaces

3.1 Resolution of indeterminacy points

Let χ : S− → S′ be a birational map of nonsingular projective surfaces. It is well-known (see [32]) that there exist birational morphisms of nonsingular surfaces σ : X → S and ϕ : X → S′ such that the following diagram is commutative

\xymatrix{ & X \ar[dl]_{σ} \ar[dr]^{ϕ} \\ S \ar@{-->}[rr] &  & S′.}  (9)

It is called a resolution of indeterminacy points of χ. Recall also that any birational morphism can be factored into a finite sequence of blow-ups of points. Let

σ : X = X_{N} ⟶^{σ_{N}} X_{N−1} ⟶^{σ_{N−1}} ⋯ ⟶^{σ_{2}} X_{1} ⟶^{σ_{1}} X_{0} = S  (10)

be such a factorization. Here σ_{i} : X_{i} → X_{i−1} is the blow-up of a point x_{i} ∈ X_{i−1}. Let

E_{i} = σ_{i}^{−1}(x_{i}), E_{i} = (σ_{i+1} ∘ … σ_{N})^{−1}(E_{i}).  (11)

Let H′ be a very ample divisor class on S′ and H′ be the corresponding complete linear system |H′|. Let H_{N} = ϕ^{∗}(H′). Define m(x_{N}) as the smallest positive number such that H_{N} + m(x_{N})E_{N} = σ_{N}^{∗}(H_{N−1}) for some linear system H_{N−1} on X_{N−1}. Then proceed inductively to define linear systems H_{k} on each X_{k} such that H_{k+1} + m(x_{k+1})E_{k+1} = σ_{k+1}^{∗}(H_{k}), and finally a linear system H = H_{0} on S such that H_{1} + m(x_{1})E_{1} = σ_{1}^{∗}(H). It follows from the definition that

ϕ^{∗}(H′) = σ^{∗}(H) − ∑_{i=1}^{N} m(x_{i})E_{i}.  (12)

The proper transform of H′ on S under χ is contained in the linear system H. It consists of curves which pass through the points x_{i} with multiplicity ≥ m_{i}. We denote it by

χ^{−1}(H′) = |H − m(x_{1})x_{1} − ⋯ − m(x_{N})x_{N}|,

where H ⊂ |H|. Here for a curve on S to pass through a point x_{i} ∈ X_{i−1} with multiplicity ≥ m(x_{i}) means that the proper transform of the curve on X_{i−1} has x_{i} as a point of multiplicity ≥ m(x_{i}). The divisors E_{i} are called the exceptional curves of the resolution σ : X → S of the birational map χ. Note that E_{i} is an irreducible curve if and only if σ_{i+1} ∘ … ∘ σ_{N} : X → X_{i} is an isomorphism over E_{i} = σ^{−1}(x_{i}).

The set of points x_{i} ∈ X_{i}, i = 1, …, N, is called the set of indeterminacy points, or base points, or fundamental points of χ. Note that, strictly speaking, only one of them, x_{1}, lies in S. However, if σ_{1} ∘ … ∘ σ_{i} : X_{i} → S is an isomorphism in a neighborhood of x_{i+1} we can identify this point with a point in S. Let {x_{i}, i ∈ I} be the set of such points. Points x_{j}, j ∉ I, are infinitely near points. A precise meaning of this classical notion is as follows.

Let S be a nonsingular projective surface and B(S) be the category of birational morphisms π : S′ → S of nonsingular projective surfaces. Recall that a morphism from (S′ →^{π′} S) to (S′′ →^{π′′} S) in this category is a regular map ϕ : S′ → S′′ such that π′′ ∘ ϕ = π′.

Definition 3.1. The bubble space S^{bb} of a nonsingular surface S is the factor set

S^{bb} = ( ⋃_{(S′→^{π′}S)∈B(S)} S′)/R,

where R is the following equivalence relation: x′ ∈ S′ is equivalent to x′′ ∈ S′′ if the rational map π′′^{−1} ∘ π′ : S′− → S′′ maps isomorphically an open neighborhood of x′ to an open neighborhood of x′′.

It is clear that for any π : S′ → S from B(S) we have an injective map i_{S′} : S′ → S^{bb}. We will identify points of S′ with their images. If ϕ : S′′ → S′ is a morphism in B(S) which is isomorphic in B(S′) to the blow-up of a point x′ ∈ S′, any point x′′ ∈ ϕ^{−1}(x′) is called infinitely near point to x′ of the first order. This is denoted by x′′ ≻ x′. By induction, one defines an infinitely near point of order k, denoted by x′′ ≻_{k} x′. This defines a partial order on S^{bb}.

We say that a point x ∈ S^{bb} is of height k, if x ≻_{k} x_{0} for some x_{0} ∈ S. This defines the height function on the bubble space

ht_{S} : S^{bb} → ℕ.

Clearly, S = ht^{−1}(0).

It follows from the known behavior of the canonical class under a blow-up that

K_{X} = σ^{∗}(K_{S}) + ∑_{i=1}^{N} E_{i}.  (13)

The intersection theory on a nonsingular surface gives

H^{′2} = (ϕ^{∗}(H′))^{2} = (σ^{∗}(H) − ∑_{i=1}^{N} m(x_{i})E_{i})^{2} = H^{2} − ∑_{i=1}^{N} m(x_{i})^{2}, (14)

K_{S′} · H′ = K_{S} · H − ∑_{i=1}^{N} m(x_{i}).

Example 3.2. Let χ : ℙ^{2} − → ℙ^{2} be a Cremona transformation, H′ = |ℓ| be the linear system of lines in ℙ^{2}, and H ⊂ |nℓ|. The formulas (14) give

n^{2} − ∑_{i=1}^{N} m(x_{i})^{2} = 1,  (15)

3n − ∑_{i=1}^{N} m(x_{i}) = 3.

The linear system H is written in this situation as H = |nℓ − ∑_{i=1}^{N} m_{i}x_{i}|. For example, a quadratic transformation with 3 base points p_{1}, p_{2}, p_{3} is given by the linear system |2ℓ − p_{1} − p_{2} − p_{3}|. In the case of the standard quadratic transformation τ_{1} the curves E_{1}, E_{2}, E_{3} are irreducible, the map σ_{1} : X_{1} → X_{0} = ℙ^{2} is an isomorphism in a neighborhood of p_{2}, p_{3} and the map σ_{2} : X_{2} → X_{1} is an isomorphism in a neighborhood of σ^{−1}(p_{3}). This shows that we can identify p_{1}, p_{2}, p_{3} with points on ℙ^{2}. In the case of the transformation (2), we have σ_{1}(p_{2}) = p_{1} and p_{3} can be identified with a point on ℙ^{2}. So in this case p_{2} ≻ p_{1}. For the transformation (3) we have p_{3} ≻ p_{2} ≻ p_{1}.

For a Geiser involution (resp. a Bertini involution) we have H = |8ℓ − 3p_{1} − ⋯ − 3p_{7}| (resp. H = |17ℓ − 6p_{1} − ⋯ − 6p_{8}|).

3.2 G-surfaces

Let G be a finite group. A G-surface is a pair (S, ρ), where S is a nonsingular projective surface and ρ is an isomorphism from G to a group of automorphisms of S. A morphism of the pairs (S, ρ) → (S′, ρ′) is defined to be a morphism of surfaces f : S → S′ such that ρ′(G′) = f ∘ρ(G)∘f^{−1}. In particular, two subgroups of Aut(S) define isomorphic G-surfaces if and only if they are conjugate inside of Aut(S). Often, if no confusion arises, we will denote a G-surface by (S, G).

Let χ : S → S′ be a birational map of G-surfaces. Then one can Gequivariantly resolve χ, in the sense that one can find the diagram (9) where all maps are morphisms of G-surfaces. The group G acts on the surface X permuting the exceptional configurations E_{i} in such a way that E_{i} ⊂ E_{j} implies g(E_{i}) ⊂ g(E_{j}). This defines an action of G on the set of indeterminacy points of χ (g(x_{i}) = x_{j} if g(E_{i}) = g(E_{j})). The action preserves the order, i.e. x_{i} ≻ x_{j} implies g(x_{i}) ≻ g(x_{j}), so the function ht : {x_{1}, …, x_{N}} → ℕ is constant on each orbit Gx_{i}.

Let H′ = |H′| be an ample linear system on S′ and

ϕ^{∗}(H′) = σ^{∗}(H) − ∑_{i=1}^{N} m(x_{i})E_{i}

be its inverse transform on X as above. Everything here is G-invariant, so H is a G-invariant linear system on S and the multiplicities m(x_{i}) are constant on the G-orbits. So we can rewrite the system in the form

ϕ^{∗}(H′) = σ^{∗}(H) − ∑_{κ∈I} m(κ)E_{κ},

where I is the set of G-orbits of indeterminacy points. For any κ ∈ I we set m(κ) = m(x_{i}), where x_{i} ∈ κ and E_{κ}, = ∑_{x_{i}∈κ} E_{i}. Similarly one can rewrite the proper transform of H′ on S

|H − ∑_{κ∈I} m(κ)κ|.  (16)

Now we can rewrite the intersection formula (14) in the form

H′^{2} = H^{2} − ∑_{κ∈I} m(κ)^{2}d(κ)  (17)
K_{S′} ∘ H′ = K_{S} · H − ∑_{κ∈I} m(κ)d(κ),

where d(κ) = #{i : x_{i} ∈ κ}.

Remark 3.3. In the arithmetical analog of the previous theory all the notations become more natural. Our maps are maps over a perfect ground field k. A blow-up is the blow-up of a closed point in the scheme-theoretical sense, not necessary k-rational. An exceptional curve is defined over k but when we replace k with its algebraic closure k̄, it splits into the union of conjugate exceptional curves over k̄. So, in the above notation, κ means a closed point on S or an infinitely near point. The analog of d(κ) is of course the degree of a point, i.e. the extension degree [k(x) : k], where k(x) is the residue field of x.

3.3 The G-equivariant bubble space

Here we recall Manin’s formalism of the theory of linear systems with basis condition in its G-equivariant form (see [45]).

First we define the G-equivariant bubble space of a G-surface S as a Gequivariant version (S, G)^{bb} of Definition 3.1. One replaces the category B(S) of birational morphisms S′ → S with the category B(S, G) of birational morphisms of G-surfaces. In this way the group G acts on the bubble space (S, G)^{bb} and the height function becomes constant on G-orbits. Let

Z^{∗}(S, G) = \underrightarrow{lim}Pic(S′),  (18)

where the inductive limit is taken with respect to the functor Pic from the category B(S, G) with values in the category of abelian groups defined by S′ → Pic(S′). The group Z^{∗}(S, G) is equipped with a natural structure of G-module. Also it is equipped with the following natural structures.

(a) A symmetric G-invariant pairing

Z^{∗}(S, G) × Z^{∗}(S, G) → ℤ

induced by the intersection pairing on each Pic(S′).

(b) A distinguished cone of effective divisors classes in Z^{∗}(S, G)

Z_{+}^{∗}(S, G) = \underrightarrow{lim}Pic_{+}(S′),

where Pic_{+}(S′) is the cone of effective divisor classes on each S′ from B(S, G).

(c) A distinguished G-equivariant homomorphism

K : Z^{∗}(S, G) → ℤ, K(z) = K_{S′} · z, for any S′ → S from B(S, G).

Let f : S′ → S be a morphism from B(S, G) and E_{1}, …, E_{n} be its exceptional curves. We have a natural splitting

Pic(S′) = f^{∗}(Pic(S)) ⊕ ℤ[E_{1}] ⊕ ⋯ ⊕ ℤ[E_{n}].

Now let Z_{0}(S, G) = ℤ^{(S,G)^{bb}} be the free abelian group generated by the set (S, G)^{bb}. Identifying exceptional curves with points in the bubble space, and passing to the limit we obtain a natural splitting

Z^{∗}(S, G) = Z_{0}(S, G) ⊕ Pic(S).  (19)

Passing to invariants we get the splitting

Z^{∗}(S, G)^{G} = Z_{0}(S, G)^{G} ⊕ Pic(S)^{G}.  (20)

Write an element of Z^{∗}(S, G)^{G} in the form

z = D + ∑_{κ∈O} m(κ)κ,

where O is the set of G-orbits in Z_{0}(S, G)^{G} and D is a G-invariant divisor class on S. Then

(a) z · z′ = D · D′ − ∑_{κ∈O} m(κ)m′(κ)d(κ);

(b) z ∈ Z_{+}^{∗}(S, G) if and only if D ∈ Pic_{+}(S)^{G}, m(κ) ≥ 0 and m(κ′) ≤ m(κ) if κ′ ≻ κ;

(c) K(z) = D · K_{S} − ∑_{κ∈O} m(κ)d(κ).

Let ϕ : S′ → S be an object of B(S, G). Then we have a natural map ϕ_{bb} : (S′, G)^{bb} → (S, G)^{bb} which induces an isomorphism ϕ_{bb}^{∗} : Z(S, G) → Z(S′, G). We also have a natural isomorphism ϕ_{∗}^{bb} : Z(S′, G) → Z(S, G). None of these maps preserves the splitting (19). Resolving indeterminacy points of any birational map χ : (S, G)− → (S′, G′) we can define

• proper direct transform map χ_{∗} : Z^{∗}(S, G) →^{∼} Z^{∗}(S′, G);

• proper inverse transform map χ^{∗} : Z^{∗}(S′, G) →^{∼} Z^{∗}(S, G).

The group Z^{∗}(S, G) equipped with all above structures is one of the main Gbirational invariants of S. It can be viewed as the Picard group of the bubble space (S, G)^{bb}.

The previous machinery gives a convenient way to consider the linear systems defining rational maps of surfaces. Thus we can rewrite (12) in the form |z|, where z = H − ∑ m_{i}x_{i} is considered as an element of Z_{+}^{∗}(S, G). The condition that |z| is homaloidal becomes equivalent to the conditions

z^{2} = H^{2} − ∑ m_{i}^{2} = H′^{2},  (21)
K(z) = H · K_{S} − ∑ m_{i} = H′ · K_{S′}.

When S = S′ = ℙ^{2} we get the equalities (15).

3.4 Minimal rational G-surfaces

Let (S, ρ) be a rational G-surface. Choose a birational map ϕ : S− → ℙ^{2}. For any g ∈ G, the map ϕ ∘ g ∘ ϕ^{−1} belongs to Cr(2). This defines an injective homomorphism

ι_{ϕ} : G → Cr(2).  (22)

Suppose (S′, ρ′) is another rational G-surface and ϕ′ : S′− → ℙ^{2} is a birational map.

The following lemma is obvious.

Lemma 3.4. The subgroups ι_{ϕ}(G) and ι_{ϕ′}(G) of Cr(2) are conjugate if and only if there exists a birational map of G-surfaces χ : S′− → S.

The lemma shows that a birational isomorphism class of G-surfaces defines a conjugacy class of subgroups of Cr(2) isomorphic to G. The next lemma shows that any conjugacy class is obtained in this way.

Lemma 3.5. Suppose G is a finite subgroup of Cr(2), then there exists a rational G-surface (S, ρ) and a birational map ϕ : S → ℙ^{2} such that, for all g ∈ G,

G = ϕ ∘ ρ(G) ∘ ϕ^{−1}.

Proof. We give two proofs. The fist one is after A. Verra. Let D = ∩_{g∈G}dom(g), where dom(g) is an open subset on which g is defined. Then U = ∩_{g∈G}g(D) is an open invariant subset of ℙ^{2} on which g ∈ G acts biregularly. Order G in some way and consider a copy of ℙ_{g}^{2} indexed by g ∈ G. For any u ∈ U let g(u) ∈ ℙ_{g}^{2}. We define a morphism

ϕ : U → ∏_{g∈G} ℙ_{g}^{2}, u ↦ (g(u))_{g∈G}.

Define an action of G on ϕ(U) by g′((x_{g})_{g∈G}) = (x_{gg′})_{g∈G}. Then ϕ is obviously G-equivariant. Now define V as the Zariski closure of ϕ(U) in the product. It is obviously a G-invariant surface which contains an open G-invariant subset G-isomorphic to U. It remains to replace V by its G-equivariant resolution of singularities (which always exists).

The second proof is standard. Let U be as above and U′ = U/G be the orbit space. It is a normal algebraic surface. Choose any normal projective completion X′ of U′. Let S′ be the normalization of X′ in the field of rational functions of U. This is a normal projective surface on which G acts by biregular transformations. It remains to define S to be a G-invariant resolution of singularities (see also [23]).

Summing up, we obtain the following result.

Theorem 3.6. There is a natural bijective correspondence between birational isomorphism classes of rational G-surfaces and conjugate classes of subgroups of Cr(2) isomorphic to G.

So our goal is to classify G-surfaces (S, ρ) up to birational isomorphism of G-surfaces.

Definition 3.7. A minimal G-surface is a G-surface (S, ρ) such that any birational morphism of G-surfaces (S, ρ) → (S′, ρ′) is an isomorphism. A group G of automorphisms of a rational surface S is called a minimal group of automorphisms if the pair (S, ρ) is minimal.

Obviously, it is enough to classify minimal rational G-surfaces up to birational isomorphism of G-surfaces.

Before we state the next fundamental result, let us recall some terminology. A conic bundle structure on a rational G-surface (S, G) is a G-equivariant

morphism ϕ : S → ℙ^{1} such that the fibres are isomorphic to a reduced conic in ℙ^{2}. A Del Pezzo surface is a surface with ample anti-canonical divisor −K_{S}.

Theorem 3.8. Let S be a minimal rational G-surface. Then either S admits a structure of a conic bundle with Pic(S)^{G} ≅ ℤ^{2}, or S is isomorphic to a Del Pezzo surface with Pic(S)^{G} ≅ ℤ.

An analogous result from the classical literature is proven by using the method of the termination of adjoints, first introduced for linear system of plane curves in the work of G. Castelnuovo. It consists in replacing a linear system |D| with the linear system |D+K_{S}| and repeat doing this to stop only if the next step leads to the empty linear system. The application of this method to finding a G-invariant linear system of curves on the plane was initiated in the works of S. Kantor [41], who essentially stated the theorem above but without the concept of minimality. In arithmetical situation this method was first applied by Enriques [28]. A first modern proof of the theorem was given by Manin [44] and by the second author [37] (an earlier proof of Manin used the assumption that G is an abelian group). Nowadays the theorem follows easily from a G-equivariant version of Mori’s theory (see [42], Example 2.18) and the proof can be found in literature ([7], [22]). For this reason we omit the proof.

Recall the classification of Del Pezzo surfaces (see [24], [45]). The number d = K_{S}^{2} is called the degree. By Noether’s formula, 1 ≤ d ≤ 9. For d ≥ 3, the anti-canonical linear system | − K_{S}| maps S onto a nonsingular surface of degree d in ℙ^{d}. If d = 9, S ≅ ℙ^{2}. If d = 8, then S ≅ ℙ^{1} × ℙ^{1}, or S ≅ F_{1}, where as always we denote by F_{n} the minimal ruled surface ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(n)). For d ≤ 7, a Del Pezzo surface S is isomorphic to the blow-up of n = 9 − d points in ℙ^{2} satisfying the following conditions

• no three are on a line;

• no six are on a conic;

• if n = 8, then then the points are not on a plane cubic which has one of them as its singular point.

For d = 2, the linear system |−K_{S}| defines a finite morphism of degree 2 from S to ℙ^{2} with a nonsingular quartic as the branch curve. Finally, for d = 1, the linear system | − 2K_{S}| defines a finite morphism of degree 2 onto a quadric cone Q ⊂ ℙ^{3} with the branch curve cut out by a cubic.

For a minimal Del Pezzo G-surface the group Pic(S)^{G} is generated by K_{S} if S is not isomorphic to ℙ^{2} or ℙ^{1} × ℙ^{1}. In the latter cases it is generated by \frac{1}{3}K_{S} or \frac{1}{2}K_{S}, respectively.

A conic bundle surface is either isomorphic to F_{n} or to a surface obtained from F_{n} by blowing up a finite set of points, no two lying in a fibre of a ruling. The number of blow-ups is equal to the number of singular fibres of the conic bundle fibration. We will exclude the surfaces F_{0} and F_{1}, considering them as Del Pezzo surfaces.

There are minimal conic bundles with ample −K_{S} (see Proposition 5.2).

4 Automorphisms of minimal ruled surfaces

4.1 Some of group theory

We employ the standard notations for groups used by group-theorists (see [18]):

C_{n}, a cyclic group of order n, n = C_{n} if no confusion arises; n^{r} = C_{n}^{r}, the direct sum of r copies of C_{n} (not to be confused with cyclic

group of order n^{r}); S_{n}, the permutation group of degree n, A_{n}, the alternating group of degree n, D_{2n}, the dihedral group of order 2n, Q_{4n} = ⟨a, b | a^{2n} = 1, b^{2} = a^{n}, b^{−1}ab = a^{−1}⟩, dicyclic group of order 4n, a

generalized quaternion group if n = 2^{k};

H_{n}(p), the Heisenberg group of unipotent n × n-matrices with entries in 𝔽_{p}.

GL(n) = GL(n, ℂ), general linear group over ℂ, SL(n) = SL(n, ℂ), special linear group over ℂ, PGL(n) = GL(n, ℂ)/ℂ^{∗}, general projective linear group over ℂ, O(n), the orthogonal linear group over ℂ, PO(n), the projective orthogonal linear group over ℂ, L_{n}(q) = PSL(n, 𝔽_{q}), where q = p^{r} is a power of a prime number p; T ≅ A_{4}, O ≅ S_{4} ≅ PGL(2, 𝔽_{3}), I ≅ A_{5} ≅ L_{2}(5) ≅ L_{2}(2^{2}), tetrahedral,

octahedron, icosahedron subgroups of PGL(2),

\overline{T} ≅ SL(2, 𝔽_{3}), Ō ≅ GL(2, 𝔽_{3}), Ī ≅ SL(2, 𝔽_{5}), \overline{D}_{2n} ≅ Q_{4n}, binary tetrahedral, binary octahedron,binary icosahedron, binary dihedral subgroups of SL(2),

A•B is an upward extension of B with help of a normal subgroup A;

A : B is a split extension A•B, i.e. a semi-direct product A ⋊ B ((it is defined by a homomorphism ϕ : B → Aut(A));

A^{•}B is a non-split extension A•B,

nA = n^{•}A, where the normal group n is equal to the center.

p^{a+b} = C_{p}^{a}•C_{p}^{b}, where p is prime;

A △ B (or A △_{D} B), the diagonal product of A and B over their common homomorphic image D (i.e. the subgroup of A × B of pairs (a, b) such that α(a) = β(b) for some surjections α : A → D, β : B → D). When D is omitted it means that D is the largest possible;

\frac{1}{m}[A × B] = A △_{D} B, where #D = m; A ≀ S_{n}, the wreath product, i.e. A^{n} : S_{n}, where S_{n} is the symmetric group

acting on A^{n} by permuting the factors; μ_{n}, the group of nth roots of unity with generator ϵ_{n} = e^{2πi/n}.

We will often use the following simple result from group theory which is known as Goursat’s Lemma.

Lemma 4.1. Let G be a finite subgroup of the product A × B of two groups A and B. Let p_{1} : A×B → A, p_{2} : A×B → B be the projection homomorphisms. Let G_{i} = p_{i}(G), H_{i} = Ker(p_{j}|G), i ≠ j = 1, 2. Then H_{i} is a normal subgroup in G_{i}. The map ϕ : G_{1}/H_{1} → G_{2}/H_{2} defined by ϕ(aH_{1}) = p_{2}(a)H_{2} is an isomorphism, and

G = G_{1} △_{D} G_{2},

where D = G_{1}/H_{1}, α : G_{1} → D is the projection map to the quotient, and β is the composition of the projection G_{2} → G_{2}/H_{2} and ϕ^{−1}.

Note some special cases:

G △_{1} G′ ≅ G × G′, G △_{G′} G′ = {(g, α(g)) ∈ G × G′, g ∈ G_{1}},

where α : G → G′ is a surjection and G′ → G′ is the identity.

We will be dealing with various group extensions. The following lemma is known in group theory as the Schur-Zassenhaus Theorem. Its proof can be found in [30], 6.2.

Lemma 4.2. Let A•B be an extension of groups. Suppose that the orders of A and B are coprime. Then the extension splits. If moreover A or B is solvable, then all subgroups of A : B defining splittings are conjugate.

We will often use the following simple facts, their proofs are left to the reader (or can be found in www.planetmath.org).

Lemma 4.3. A subgroup of D_{2n} = ⟨a, b | a^{n} = b^{2} = b^{−1}aba = 1⟩ is either cyclic or dihedral. A normal subgroup H is either cyclic subgroup ⟨a⟩, or n = 2k and H is one of the following two subgroups ⟨a^{2}, b⟩, ⟨a^{2}, ab⟩ of index 2. These two subgroups are interchanged under the outer automorphism a ↦ a, b ↦ ab. If H is cyclic of order n/k, the quotient group is isomorphic to D_{2n/k}.

The group of Aut(D_{2n}) is isomorphic to (ℤ/n)^{∗} : n and it is generated by the transformations a ↦ a^{s}, b ↦ a^{t}b. The subgroup of interior automorphisms is generated by transformations a ↦ a^{−1}, b ↦ b and a ↦ a, b ↦ a^{2}b.

It will be convenient to list all isomorphism classes of non-abelian groups of order 16.

Notation Center  LCS  Extensions  Presentation
2 × D_{8}  2^{2} 16, 2, 1 2^{1+3}, 2^{2+2}, 2^{3+1},  a^{4} = b^{2} = c^{2} = 1,
(2 × 4) : 2 [a, b]a^{2} = [a, c] = [b, c] = 1
2 × Q_{8} 2^{2} 16, 2, 1 2^{2+2}, a^{4} = a^{2}b^{−2} = a^{2}[a, b] = 1
(2 × 4)^{•}2
D_{16} 2 16, 4, 2, 1 8 : 2, 2D_{8}, a^{8} = b^{2} = a^{2}[a, b] = 1
(2^{2})^{•}4, D_{8}^{•}2
SD_{16} 2 16, 4, 2, 1 8 : 2, D_{8}^{•}2, a^{8} = b^{2} = [a, b]a^{−2} = 1
2D_{8}, (2^{2})^{•}4
Q_{16} 2 16, 4, 2, 1 8^{•}2, 2D_{8}, a^{8} = a^{4}b^{−2} = [a, b]a^{2} = 1
4^{•}(2^{2})
AS_{16} 4 16, 2, 1 2^{1+3}, D_{8} : 2 a^{4} = b^{2} = c^{2} = [a, b] = 1
4(2^{2}), (2 × 4) : 2  [c, b]a^{−2} = [c, a] = 1
K_{16} 2^{2} 16, 2, 1 2^{2+2}, (2 × 4)^{•}2 a^{4} = b^{4} = [a, b]a^{2} = 1
2^{•}(2 × 4), 4 : 4
L_{16} 2^{2} 16, 2, 1 2^{2} : 4, 2^{•}(2 × 4) a^{4} = b^{2} = c^{2} = 1,
(2 × 4) : 2 [c, a]b = [a, b] = [c, b] = 1
M_{16} 4 16, 2, 1 8 : 2, 4(2^{2}) a^{8} = b^{2} = 1, [a, b]a^{4} = 1
2^{•}(2 × 4)

Table 1. Non-abelian groups of order 16

Recall that there are two non-isomorphic non-abelian groups of order 8, D_{8} and Q_{8}.

Finally we describe central extension of polyhedral and binary polyhedral groups. Recall that the isomorphism classes of central extensions A•G, where A is an abelian group, are parametrized by the 2-cohomology group H^{2}(G, A). We will assume that A ≅ p, where p is prime. We will use the following facts about the cohomology groups of polyhedral and binary polyhedral groups which can be found in text-books on group cohomology (see, for example, [1]).

Lemma 4.4. Let G be a polyhedral group or a binary polyhedral group. If G ≅ n is cyclic, then H^{2}(G, p) ≅ p if p | n and zero otherwise. If G is not cyclic, then H^{2}(G, p) = 0 if p ≠ 2, 3. Moreover

(i) If G is a polyhedral group, then

H^{2}(G, 2) ≅ \begin{cases}2 & if G ≅ D_{2n}, n odd, \\ 2^{3} & if G ≅ D_{2n}, n even, \\ 2 & if G ≅ T, \\ 2^{2} & if G ≅ O, \\ 2 & if G ≅ I. \end{cases}

H^{2}(G, 3) ≅ \begin{cases}3 & if G ≅ T, \\ 1 & otherwise. \end{cases}

(ii)If G is a binary polyhedral group, then

H^{2}(G, 2) ≅ \begin{cases}2 & if G ≅ \overline{D}_{2n}, n odd, \\ 2^{2} & if G ≅ \overline{D}_{2n}, n even, \\ 2 & if G ≅ \overline{O}, \\ 1 & otherwise. \end{cases}

H^{2}(G, 3) ≅ \begin{cases}3 & if G ≅ \overline{T} , \\ 1 & otherwise. \end{cases}

4.2 Finite groups of projective automorphisms

We start with the case S = ℙ^{2}, where Aut(S) ≅ PGL(3). To save space we will often denote a projective transformation

(x_{0}, x_{1}, x_{2}) ↦ (L_{0}(x_{0}, x_{1}, x_{2}), L_{1}(x_{0}, x_{1}, x_{2}), L_{2}(x_{0}, x_{1}, x_{2}))

by [L_{0}(x_{0}, x_{1}, x_{2}), L_{1}(x_{0}, x_{1}, x_{2}), L_{2}(x_{0}, x_{1}, x_{2})].

Recall some standard terminology from the theory of linear groups. Let G be a subgroup of the general linear group GL(V) of a complex vector space V. The group G is called intransitive if the representation of G in V is reducible. Otherwise it is called transitive. A transitive group G is called imprimitive if it contains an intransitive normal subgroup G′. In this case V decomposes into a direct sum of G′-invariant proper subspaces, and elements from G permute them. A group is primitive if it is neither intransitive, nor imprimitive. We reserve this terminology for subgroups of PGL(V) keeping in mind that each such group can be represented by a subgroup of GL(V).

Let G′ be a finite intransitive subgroup of GL(3) and G be its image in PGL(3). Then G′ is conjugate to a subgroup ℂ^{∗} × GL(2) of block matrices.

To classify such subgroups we have to classify subgroups of GL(2). We will use the well-known classification of finite subgroups of PGL(2). They are isomorphic to one of the following polyhedral groups

• a cyclic group C_{n};

• a dihedral group D_{2n} of order 2n ≥ 2; • the tetrahedral group T ≅ A_{4} of order 12;

• the octahedron group O ≅ S_{4} of order 24;

• the icosahedron group I ≅ A_{5} of order 60.

Two isomorphic subgroups are conjugate subgroups of PGL(2).

The pre-image of such group in SL(2, ℂ) under the natural map

SL(2) → PSL(2) = SL(2)/(±1) ≅ PGL(2)

is a double extension \overline{G} = 2•G. The group \overline{G} = 2•G is called a binary polyhedral group. A cyclic group of odd order is isomorphic to a subgroup SL(2) intersecting trivially the center.

Consider the natural surjective homomorphism of groups

β : ℂ^{∗} × SL(2) → GL(2), (c, A) ↦ cA.

Its kernel is the subgroup {(1, I_{2}), (−1, −I_{2})}.

Let G be a finite subgroup of GL(2) with center Z(G). Since cA = (−c)(−A) and det(cA) = c^{2} det A, we see that G̃ = β^{−1}(G) is a subgroup of μ_{2m} × Ḡ′, where \overline{G′} is a binary polyhedral group with nontrivial center whose image G′ in PGL(2) is isomorphic to G/Z(G). The homomorphism β defines an isomorphism from the kernel H_{2} of the first projection G̃ → μ_{2m} onto the subgroup G_{0} = Ker(det : G → ℂ^{∗}). Also it defines an isomorphism from the kernel H_{1} of the second projection G̃ → Ḡ′ onto Z(G). Applying Lemma 4.1, we obtain

G̃ ≅ μ_{2m} △_{D} \overline{G}′, D = Ḡ′/G_{0}.

Lemma 4.5. Let G be a finite non-abelian subgroup of GL(2). Then G = β(G̃), where G̃ ⊂ ℂ^{∗} × SL(2, ℂ) is conjugate to one of the following groups

(i) G̃ = μ_{2m} × Ī, G ≅ m × Ī; (ii)G̃ = μ_{2m} × \overline{O}, G ≅ m × \overline{O}; (iii)G̃ = μ_{2m} × \overline{T} , G ≅ m × \overline{T}; (iv)G̃ = μ_{2m} × Q_{4n}, G ≅ m × Q_{4n};

(v) G̃ = \frac{1}{2}[μ_{4m} × \overline{O}], G ≅ 2m•O ≅ (m × \overline{T})•2 (split if m = 1, 2); (vi)G̃ = \frac{1}{3}[μ_{6m} × \overline{T}], G ≅ 2m•T ≅ (m × 2^{2})•3 (split if m = 1, 3); (vii)G̃ ≅ \frac{1}{2}[μ_{4m} × Q_{8n}], G ≅ 2m•D_{4n} ≅ (m × Q_{4n})•2 (split if m = 1, 2); (viii) G̃ = \frac{1}{2}[μ_{4m} × Q_{4n}], G ≅ 2m•D_{2n} ≅ (m × 2n)•2 (split if m = 1, 2); (ix)G̃ = \frac{1}{4}[μ_{4m} × Q_{4n}], n is odd, G ≅ m•D_{2n} ≅ (m × n)•2 (split if m = 1, 2).

Note that, although Q_{8n} has two different non-cyclic subgroups of index 2, they are conjugate under an element of SL(2), so they lead to conjugate subgroups in GL(2).

Lemma 4.4 gives us some information when some of these extensions split.

An abelian subgroup G ⊂ GL(2) is conjugate to a subgroup of diagonal matrices of the form (ϵ_{m}^{a}, ϵ_{n}^{b}), where ϵ_{m}, ϵ_{n} are primitive roots of unity and a, b ∈ ℤ. Let d = (m, n), m = du, n = dv, d = kq for some fixed positive integer k. Let H_{1} = ⟨ϵ_{m}^{k}⟩ ⊂ ⟨ϵ_{m}⟩, H_{2} = ⟨ϵ_{n}^{k}⟩ ⊂ ⟨ϵ_{n}⟩ be cyclic subgroups of index k. Applying Lemma 4.1 we obtain

G ≅ ⟨ϵ_{m}⟩ △_{k} ⟨ϵ_{n}⟩

where the homomorphisms ⟨ϵ_{m}⟩ → k, ⟨ϵ_{n}⟩ → k differ by an automorphism of the cyclic group ⟨ϵ_{k}⟩ ≅ k defined by a choice of a new generator ϵ_{m}^{s}, (s, k) = 1. In this case

G = (⟨ϵ_{m}^{k}⟩ × ⟨ϵ_{n}^{k}⟩_{•}⟨ϵ_{k}⟩  (23)

is of order mn/k = uvkq^{2}. In other words, G consists of diagonal matrices of the form (ϵ_{m}^{a}, ϵ_{n}^{b}), where a ≡ sb mod k.

Corollary 4.6. Let G be an intransitive finite subgroup of GL(3). Then its image in PGL(3) consists of transformations [ax_{0} + bx_{1}, cx_{0} + dx_{1}, x_{2}], where the matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) form a non-abelian finite subgroup H of GL(2) from Lemma 4.5 or an abelian group of the form (23).

Now suppose G is transitive but imprimitive subgroup of PGL(3). Let G′ be its largest intransitive normal subgroup. Then G/G′ permutes transitively the invariant subspaces of G′, hence we may assume that all of them are onedimensional. Replacing G by a conjugate group we may assume that G′ is a subgroup of diagonal matrices. We can represent its elements by diagonal matrices g = (ϵ_{m}^{a}, ϵ_{n}^{b}, 1), where a ≡ sb mod k as in (23). The group G contains a cyclic permutation τ of coordinates. Since G′ is a normal subgroup of G, we get τ^{−1}gτ = (ϵ_{n}^{−b}, ϵ_{n}^{−b}ϵ_{m}^{a}, 1) ∈ G′. This implies that n|bm, m|an, hence u|b, v|a. Since (ϵ_{m}, ϵ_{n}^{s}, 1) or (ϵ_{m}^{s′}, ϵ_{n}, 1), ss′ ≡ 1 mod k, belongs to G we must have u = v = 1, i.e. m = n = d. Therefore G′ consists of diagonal matrices g = (ϵ_{d}^{a}, ϵ_{d}^{sa}, 1). Since τ^{−1}gτ = (ϵ_{d}^{−sa}, ϵ_{d}^{a−sa}, 1) ∈ G′, we get a − sa ≡ −s^{2}a mod k for all a ∈ ℤ/mℤ. Hence the integers s satisfy the congruence s^{2} − s + 1 ≡ 0 mod k. If, moreover, G/G′ ≅ S_{3}, then we have an additional condition s^{2} ≡ 1 mod k, and hence either k = 1 and G′ = μ_{n} × μ_{n} or k = 3, s = 2 and G′ = n × n/k.

This gives the following.

Theorem 4.7. Let G be a transitive imprimitive finite subgroup of PGL(3). Then G is conjugate to one of the following groups

• G ≅ n^{2} : 3 generated by transformations

[ϵ_{n}x_{0}, x_{1}, x_{2}], [x_{0}, ϵ_{n}x_{1}, x_{2}], [x_{2}, x_{0}, x_{1}];

• G ≅ n^{2} : S_{3} generated by transformations

[ϵ_{n}x_{0}, x_{1}, x_{2}], [x_{0}, ϵ_{n}x_{1}, x_{2}], [x_{0}, x_{2}, x_{1}], [x_{2}, x_{0}, x_{1}];

• G = G_{n,k,s} ≅ (n × \frac{n}{k}) : 3, where k > 1, k|n and s^{2} − s + 1 = 0 mod k. It is generated by transformations

[ϵ_{n/k}x_{0}, x_{1}, x_{2}], [ϵ_{n}^{s}x_{0}, ϵ_{n}x_{1}, x_{2}], [x_{2}, x_{0}, x_{1}].

• G ≃ (n × \frac{n}{3}) : S_{3} generated by transformations

[ϵ_{n/3}x_{0}, x_{1}, x_{2}], [ϵ_{n}^{2}x_{0}, ϵ_{n}x_{1}, x_{2}], [x_{0}, x_{2}, x_{1}], [x_{1}, x_{0}, x_{2}].

The next theorem is a well-known result of Blichfeldt [10].

Theorem 4.8. Any primitive finite subgroup G of PGL(3) is conjugate to one of the following groups.

1. The icosahedron group A_{5} isomorphic to L_{2}(5). It leaves invariant a nonsingular conic.

2. The Hessian group of order 216 isomorphic to 3^{2} : L_{2}(3). It is realized as the group of automorphisms of the Hesse pencil of cubics

x^{3} + y^{3} + z^{3} + txyz = 0.

The subgroup L_{2}(3) is isomorphic to 2•A_{4} ≅ \overline{T} and permutes the four reducible members of the pencil.

3. The Klein group of order 168 isomorphic to L_{2}(7) (realized as the full group of automorphisms of the Klein quartic x^{3}y + y^{3}z + z^{3}x = 0).

4. The Valentiner group of order 360 isomorphic to A_{6}. It can be realized as the full group of automorphisms of the nonsingular plane sextic

10x^{3}y^{3} + 9zx^{5} + y^{5} − 45x^{2}y^{2}z^{2} − 135xyz^{4} + 27z^{6} = 0.

5. Subgroups of the Hessian group:

• 3^{2} : 4;

• 3^{2} : Q_{8}.

4.3 Finite groups of automorphisms of F_{0}

Since F_{0} is isomorphic to a nonsingular quadric in ℙ^{3}, the group Aut(F_{0}) is isomorphic to the projective orthogonal group PO(4). The classification of finite subgroups of O(4) is due to É. Goursat [31] (in the real case see a modern account in [19]). Goursat’ Lemma 4.1 plays an important role in this classification.

Obviously,

Aut(F_{0}) ≅ PGL(2) ≀ S_{2},

First we classify subgroups of PGL(2) × PGL(2) by applying Goursat’s Lemma.

Observe the following special subgroups of PGL(2) × PGL(2).

1. G = G_{1} × G_{2} is the product subgroup.

2. G △_{1} G = {(g_{1}, g_{2}) ∈ G × G : α(g_{1}) = g_{2}} ≅ G is a α-twisted diagonal subgroup. If α = id_{G}, we get the diagonal subgroup.

Note that α-twisted diagonal groups are conjugate in Aut(F_{0}) if α(g) = xgx^{−1}for some x in the normalizer of G inside Aut(ℙ^{1}). In particular, we may always assume that α is an exterior automorphism of G.

We will use the notation [p_{1}, …, p_{r}] for the Coxeter group defined by the Coxeter diagram

•  •  •  •  • p_{1}  p_{2}  p_{r} ⋯

Following [19] we write [p_{1}, …, p_{r}]^{+} to denote the index 2 subgroup of even length words in standard generators of the Coxeter group. If exactly one of the numbers p_{1}, …, p_{r} is even, say p_{k}, there are two other subgroups of index 2 denoted by [p_{1}, …, p_{r}^{+}] (resp. [^{+}p_{1}, …, p_{r}]). They consist of words which contain each generator R_{1}, …, R_{k−1} (resp. R_{k+1}, …, R_{r}) even number of times. The intersection of these two subgroups is denoted by [^{+}p_{1}, …, p_{r}^{+}]. For example,

D_{2n} = [n], T = [3, 3]^{+}, O = [3, 4]^{+}, I = [3, 5]^{+}.

Recall that each group [p_{1}, …, p_{r}] has a natural linear representation in ℝ^{r} as a reflection group. If r = 3, the corresponding representation defines a subgroup of PO(4). If r = 2, it defines a subgroup of PO(3) which acts diagonally on ℙ^{2} × ℙ^{2} and on ℙ^{1} × ℙ^{1} embedded in ℙ^{2} × ℙ^{2} by the product of the Veronese maps. We denote \overline{[p_{1}, …, p_{r}]} the quotient of [p_{1}, …, p_{r}] by its center. Similar notation is used for the even subgroups of [p_{1}, …, p_{r}].

Theorem 4.9. Let G be a finite subgroup of PGL(2) × PGL(2) not conjugate to the product A × B of subgroups of PGL(2). Then G is conjugate to one of the following groups or its image under the switching of the factors.

• \frac{1}{60}[I × I] ≅ I ≅ [3, 5]^{+};

• \frac{1}{60}[I × I] ≅ I ≅ [3, 3, 3]^{+};

• \frac{1}{24}[O × O] ≅ O ≅ [3, 4]^{+};

• \frac{1}{24}[O × O] ≅ O ≅ [2, 3, 3]^{+};

• \frac{1}{12}[T × T] ≅ T ≅ [3, 3]^{+};

• \frac{1}{2}[O × O] ≅ (T × T) : 2 ≅ \overline{[3, 4, 3]^{+}};

• \frac{1}{6}[O × O] ≅ 2^{4} : S_{3} ≅ \overline{[3, 3, 4]^{+}};

• \frac{1}{3}[T × T] ≅ 2^{4} : 3 ≅ \overline{[^{+}3, 3, 4^{+}]};

• \frac{1}{2}[D_{2m} × D_{4n}] ≅ (m × D_{2n})^{•}2 (m, n ≥ 2);

• \frac{1}{4}[D_{4m} × D_{4n}] ≅ (m × n) : 4 (m, n odd);

• \frac{1}{2k} [D_{2mk} × D_{2nk}]_{s} ≅ (m × n) : D_{2k}, (s, k) = 1);

• \frac{1}{2k} [D_{2mk} × D_{2nk}]_{s} ≅ (m × n) : D_{2k}, (s, 2k) = 1, m, n odd); • \frac{1}{k} [C_{mk} × C_{nk}]_{s} ≅ (m × n)•k ((s, k) = 1);

• \frac{1}{k} [C_{mk} × C_{nk}]_{s} ≅ (m × n)•k ((s, 2k) = 1, m, n odd;);

• \frac{1}{2}[D_{2m} × O] ≅ (m × T) : 2;

• \frac{1}{2}[D_{4m} × O] ≅ (D_{2m} × T) : 2(m ≥ 2);

• \frac{1}{6}[D_{6n} × O] ≅ (m × 2^{2}) : S_{3}(m ≥ 2);

• \frac{1}{2}[C_{2m} × O] ≅ (m × T)•2 (split if m = 1);

• \frac{1}{3}[C_{3m} × T] ≅ (m × 2^{2})•3 (split if m = 1);

• \frac{1}{2}[D_{4m} × D_{4n}] ≅ (D_{2m} × D_{2n})^{•}2 (m, n ≥ 2);

• \frac{1}{2}[C_{2m} × D_{4n}] ≅ (m × D_{2n})^{•}2 (n ≥ 2);

• \frac{1}{2}[C_{2m} × D_{2n}] ≅ (m × n) : 2 ≅ m•D_{2n}.

All other finite subgroups of Aut(ℙ^{1} × ℙ^{1}) are conjugate to a group G^{0}•2, where the quotient 2 is represented by an automorphism which interchanges the two rulings of F_{0}. It is equal to τ ∘ g, where τ is the switch (x, y) ↦ (y, x) and g ∈ PGL(2) × PGL(2).

4.4 Finite groups of automorphisms of F_{n}, n ≠ 0

Let S be a minimal ruled surface F_{n}, n ≠ 0. If n = 1, the group Aut(F_{1}) is isomorphic to a subgroup of Aut(ℙ^{2}) leaving one point fixed. We will not be interested in such subgroups so we assume that n ≥ 2.

Theorem 4.10. Let S = F_{n}, n ≠ 2. We have

Aut(F_{n}) ≅ ℂ^{n+1} : (GL(2)/μ_{n}),

where GL(2)/μ_{n} acts on ℂ^{n} by means of its natural linear representation in the space of binary forms of degree n. Moreover,

GL(2)/μ_{n} ≅ \begin{cases}ℂ^{∗} : PSL(2), & if n is even, \\ ℂ^{∗} : SL(2), & if n is odd. \end{cases}

Proof. This is of course well-known. We identify 𝔽_{n} with the weighted projective plane ℙ(1, 1, n). An automorphism is given by the formula

(t_{0}, t_{1}, t_{2}) ↦ (at_{0} + bt_{1}, ct_{0} + dt_{1}, et_{2} + f_{n}(t_{0}, t_{1})),

where f_{n} is a homogeneous polynomial of degree n. The vector space ℂ^{n+1} is identified with the normal subgroup of transformations [t_{0}, t_{1}, t_{2} + f_{n}(t_{0}, t_{1})]. The quotient by this subgroup is isomorphic to the subgroup of transformations [at_{0} + bt_{1}, ct_{0} + dt_{1}, et_{2}] modulo transformations of the form [ct_{0}, ct_{1}, c^{n}t_{2}]. This group is obviously isomorphic to GL(2)/μ_{n}. Consider the natural projection GL(2)/μ_{n} → PGL(2) ≅ PSL(2). Define a homomorphism SL(2) → GL(2)/μ_{n} by assigning to a matrix A the coset of A modulo μ_{n}. If n is even, the kernel of this homomorphism is ⟨−I_{2}⟩, so we have a splitting GL(2) ≅ ℂ^{∗} : PGL(2). If n is odd, the homomorphism is injective and defines a splitting GL(2)/μ_{n} ≅ ℂ^{∗} : SL(2).

Suppose G is a finite subgroup of Aut(𝔽_{n}). Obviously, G is contained in the subgroup GL(2)/μ_{n}.

Suppose G ∩ ℂ^{∗} = {1}. Then G is isomorphic to a subgroup of PGL(2) (resp. SL(2)) over which the extension splits. Note that the kernel ℂ^{∗} of the projections GL(2)/μ_{n} → PGL(2) or GL(2)/μ_{n} → SL(2) is the center. Thus each finite subgroup H of PGL(2) (resp. SL(2)) defines k conjugacy classes of subgroups isomorphic to H, where k = #Hom(H, ℂ^{∗}) = #G/[G, G].

If G ≅ ℂ^{∗} ≅ μ_{m} is non-trivial, the group is a central extension m•H, where H is a polyhedral group, if n is even, and a binary polyhedral group otherwise. We can apply Lemma 4.4 to find some cases when the extension must split. In other cases the structure of the group is determined by using Theorem 4.9. We leave this to the reader.

5 Automorphisms of conic bundles

5.1 Geometry of conic bundles

Let π : S → ℙ^{1} be a conic bundle with singular fibres over points in a finite set Σ ⊂ ℙ^{1}. We assume that k = #Σ > 0. Recall that each singular fibre F_{x}, x ∈ Σ, is the bouquet of two ℙ^{1}’s.

Let E be a section of the conic bundle fibration π. The Picard group of S is freely generated by the divisor classes of E, the class F of a fibre, and the classes of k components of singular fibres, no two in the same fibre. The next lemma follows easily from the intersection theory on S.

Lemma 5.1. Let E and E′ be two sections with negative self-intersection −n. Let r be the number of components of singular fibres which intersect both E and E′. Then k − r is even and

2E · E′ = k − 2n − r.

In particular,

k ≥ 2n + r.

Since a conic bundle S is isomorphic to a blow-up of a minimal ruled surface, it always contains a section E with negative self-intersection −n. If n ≥ 2, we obviously get k ≥ 4. If n = −1, since (S, g) is minimal, there exists g ∈ G such that g(E) ≠ E and E ∩ g(E) ≠ ∅. Applying the previous lemma we get

k ≥ 4.

5.2 Exceptional conic bundles

We give three different constructions of the same conic bundle, which we will call an exceptional conic bundle.

First construction.

Choose a ruling p : F_{0} → ℙ^{1} on F_{0} and fix two points on the base, say 0 and ∞. Let F_{0} and F_{∞} be the corresponding fibres. Take g + 1 points a_{1}, …, a_{g+1} on F_{0} and g +1 points a_{g+2}, …, a_{2g+2} on F_{∞} such that no two lie in the same fibre of the second ruling q : F_{0} → ℙ^{1}. Let σ : S → F_{0} be the blow-up of the points a_{1}, …, a_{2g+2}. The composition π = q ∘ σ : S → ℙ^{1} is a conic bundle with 2g +2 singular fibres R_{i} +R′_{i} over the points x_{i} = q(a_{i}), i = 1, …, 2g +2. For i = 1, …, g + 1, R_{i} = σ^{−1}(a_{i}) and R_{n+i} is the proper transform of the fibre q^{−1}(a_{i}). Similarly, for i = 1, …, n, R′_{i} is the proper transform of the fibre q^{−1}(a_{i}) and R′_{g+1+i} = σ^{−1}(a_{g+1+i}).

Let E_{0}, E_{∞} be the proper transforms of F_{0}, F_{∞} on S. Each is a section of the conic bundle π. The section E_{0} intersects R_{1}, …, R_{2g+2}, and the section E_{∞} intersects R′_{1}, …, R′_{2g+2}.

Let

D_{0} = 2E_{0} + ∑_{i=1}^{2g+2} R_{i}, D_{∞} = 2E_{∞} + ∑_{i=1}^{2g+2} R′_{i}.

It is easy to check that D_{0} ∼ D_{∞}. Consider the pencil P spanned by the curves D_{0} and D_{∞}. It has 2g + 2 simple base points p_{i} = R_{i} ∩ R′_{i}. Its general member is a nonsingular curve C. In fact, a standard formula for computing the Euler characteristic of a fibred surface in terms of the Euler characteristics of fibres shows that all members except D_{0} and D_{∞} are nonsingular curves. Let F be a fibre of the conic bundle. Since C · F = 2, the linear system |F | cuts out a g_{2}^{1} on C, so it is a hyperelliptic curve or the genus g of C is 0 or 1. The points p_{i} are obviously the ramification points of the g_{2}^{1}. Computing the genus of C we find that it is equal to g, thus p_{1}, …, p_{2g+2} is the set of ramification points. Obviously all nonsingular members are isomorphic curves. Let σ : S′ → S be the blow-up the base points p_{1}, …, p_{2g+2} and let \overline{D} denote the proper transform of a curve on S. We have

2Ē_{0} + 2Ē_{∞} + ∑_{i=1}^{2g+2}(R̄_{i} + R̄′_{i} + 2σ^{−1}(p_{i})) ∼ 2σ^{∗}(C).

This shows that there exists a double over X′ → S′ branched along the divisor ∑_{i=1}^{2g+2}(R̄_{i}+R̄′_{i}). Since R̄_{i}^{2} = R̄′_{i}^{2} = −2, the ramification divisor on X′ consists of 4g + 4 (−1)-curves. Blowing them down we obtain a surface X isomorphic to the product C × ℙ^{1}. This gives us the following.

Second construction. A pair (C, h) consisting of a nonsingular curve and an involution h ∈ Aut(C) with quotient ℙ^{1} will be called a hyperelliptic curve. If C is of genus g ≥ 2, then C is a hyperelliptic curve and h is its involution defined by the unique g_{2}^{1} on C. Let δ be an involution of ℙ^{1} defined by (t_{0}, t_{1}) ↦ (t_{0}, −t_{1}). Consider the involution τ = h × δ of the product X = C × ℙ^{1}. Its fixed points are 4g + 4 points c_{i} × {0} and c_{i} × {∞}, where X^{⟨h⟩} = {c_{1}, …, c_{2g+2}}. Let X′ be a minimal resolution of X/(τ). It is easy to see that the images of the curves {c_{i}} × ℙ^{1} are (−1)-curves on X′. Blowing them down we obtain our exceptional conic bundle.

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; C × ∞; C × {0}; X; ℙ^{1}; ℙ^{1}; ℙ^{1}; S; ℙ^{1}; ℙ^{1}; ℙ^{1}; X′; X/⟨τ⟩; −1; −1; −1; −1; −1; −1; −1; −1; −2; −2; −2; −2; −2; −2; −2; −2; −1; −1; −1; −1; ⟶; ⟵; ↑]

Third construction.

Let us consider a quasi-smooth hypersurface Y of degree 2g+2 in weighted projective space ℙ = ℙ(1, 1, g + 1, g + 1) given by an equation

F_{2g+2}(T_{0}, T_{1}) + T_{2}T_{3} = 0,  (24)

where F_{2g+2}(T_{0}, T_{1}) is a homogeneous polynomial of degree 2g + 2 without multiple roots. The surface is a double cover of ℙ(1, 1, g + 1) (the cone over a Veronese curve of degree g+1) branched over the curve F_{2g+2}(T_{0}, T_{1})+T_{2}^{2} = 0. The pre-images of the singular point of ℙ(1, 1, g + 1) with coordinate (0, 0, 1) is a pair of singular points of Y with coordinates (0, 0, 1, 0) and (0, 0, 0, 1). The singularities are locally isomorphic to the singular points of a cone of the Veronese surface of degree g + 1. Let S be a minimal resolution of Y. The pre-images of the singular points are disjoint smooth rational curves E and E′ with self-intersection −(g + 1). The projection ℙ(1, 1, g + 1, g + 1) → ℙ^{1}, (t_{0}, t_{1}, t_{2}, t_{3}) ↦ (t_{0}, t_{1}) lifts to a conic bundle on S with sections E, E′. The pencil λT_{2} + μT_{3} = 0 cuts out a pencil of curves on Y which lifts to a pencil of bisections of the conic bundle S with 2g + 2 base points (t_{0}, t_{1}, 0, 0), where F_{2g+2}(t_{0}, t_{1}) = 0.

It is easy to see that this is a general example of an exceptional conic bundle. In Construction 2, we blow down the sections E_{0}, E_{∞} to singular points. Then consider an involution g_{0} of the surface which is a descent of the automorphism of the product C × ℙ^{1} given by id_{C} × ψ, where ψ : (t_{0}, t_{1}) ↦ (t_{1}, t_{0}). The quotient by (g_{0}) gives ℙ(1, 1, g +1) and the ramification divisor is the image on S of the curve C × (1, 1) or C × (1, −1). On one of these curves g_{0} acts identically, on the other one it acts as the involution defined by the g_{2}^{1}.

Proposition 5.2. Let ϕ : S → ℙ^{1} be a minimal conic G-bundle with k ≤ 5 singular fibres. Then S is a Del Pezzo surface, unless k = 4 and S is an exceptional conic bundle.

Proof. Since k ≤ 5 we have K_{S}^{2} = 8 − k ≥ 3. By Riemann-Roch, | − K_{S}| ≠ ∅. Suppose S is not a Del Pezzo surface. Then there exists an irreducible curve C such that −K_{S} ·C ≤ 0. Suppose, the equality takes place. By Hodge’s Index Theorem, C^{2} < 0, and by the adjunction formula, C^{2} = −2 and C ≅ ℙ^{1}. If the strict inequality takes place, then C is a component of a divisor D ∈ | − K_{S}|, hence | − K_{S} − C| ≠ ∅ and |K_{S} + C| = ∅. Moreover, since K_{S}^{2} > 0, we have C ∉ | − K_{S}|. Applying Riemann-Roch to the divisor K_{S} + C we easily obtain that C is of arithmetic genus 0, and hence C ≅ ℙ^{1}. By adjunction, C^{2} ≤ −2. In both cases we have a smooth rational curve with C^{2} ≤ −2.

If k = 4 and S is an exceptional conic bundle, then S is not a Del Pezzo surface since it has sections with self-intersection −2. Assume this is not the case. Let C be the union of smooth rational curves with self-intersection < −2. It is obviously a G-invariant curve, so we can write C ∼ −aK_{S} −bf, where f is the divisor class of a fibre of ϕ. Intersecting with f we get a > 0. Intersecting with K_{S}, we get 2b > ad, where d = 8 − k ≥ 3. It follows from Lemma 5.1, that S contains a section E with self-intersection −2 or −1. Intersecting C with E we get 0 ≤ C · E = a(−K_{S} · E) − b ≤ a − b. This contradicts the previous inequality. Now let us take C to be the union of (−2)-curves. Similarly, we get 2b = ad and C^{2} = −aK_{S} · C − bC · f = −bC · f = −2ab. Let r be the number of irreducible components of C. We have 2a = C · f ≥ r and −2r ≤ C^{2} = −2ab ≤ −br. If b = 2, we have the equality everywhere, hence C consists of r = 2a disjoint sections, and 8 = rd. Since d ≥ 3, the only solution is d = 4, r = 2, and this leads to the exceptional conic bundle. Assume b = 1. Since C^{2} = −2a is even, a is a positive integer, and we get 2 = ad. Since d ≥ 3, this is impossible.

5.3 Automorphisms of an exceptional conic bundle

Let us describe the automorphism group of an exceptional conic bundle. The easiest way to do it to use Construction 3. We denote by Y_{g} an exceptional conic bundle given by equation (24). Since we are interested only in minimal groups we assume that g ≥ 1.

Since K_{Y_{g}} = O_{ℙ}(−2), any automorphism σ of Y_{g} is a restriction of an automorphism of ℙ. Let G_{1} be the subgroup of SL(2) of transformations preserving the zero divisor of F_{2g+2}(T_{0}, T_{1}) and χ_{1} : G_{1} → ℂ^{∗} be the multiplicative character of G_{1} defined by σ_{1}^{∗}(F_{2g+2}) = χ_{1}(σ_{1})F_{2g+2}. Similarly, let G_{2} be the subgroup of GL(2) of matrices preserving the zeroes of T_{2}T_{3} and let χ_{2} : G_{2} → ℂ^{∗} be the character defined by σ^{∗}(T_{2}T_{3}) = χ_{2}(σ_{2})T_{2}T_{3}. Let

(G_{1} × G_{2})^{0} = {(σ_{1}, σ_{2}) ∈ G_{1} × G_{2} : χ_{1}(σ_{1}) = χ_{2}(σ_{2})}.

In the notation of the diagonal products,

(G_{1} × G_{2})^{0} = \frac{1}{m}[G_{1} × G_{2}],

where χ_{1}(G_{1}) = μ_{m} ⊂ ℂ^{∗}. The subgroup

K = ⟨(−I_{2}, (−1)^{g+1}I_{2})⟩

acts identically on Y_{g} and the quotient group is isomorphic to Aut(Y_{g}).

Let Aut(Y_{g}) ≅ (G_{1} × G_{2})^{0}/K → PGL(2) be the homomorphism induced by the projection of G_{1} to PGL(2). Its image is a finite subgroup P of PGL(2). Its kernel H consists of cosets modulo K of pairs (±I_{2}, σ_{2}), where χ_{2}(σ_{2}) = 1. Clearly, H ≅ Ker(χ_{2}).

It is easy to see that Ker(χ_{2}) ≅ ℂ^{∗} : 2 is generated by diagonal matrices with determinant 1 and the matrix which switches the coordinates. Inside of GL(2) it is conjugate to the normalizer N of the maximal torus in SL(2). So we obtain an isomorphism

Aut(Y_{g}) ≅ N•P .  (25)

Suppose there exists a homomorphism η : G_{1} → ℂ^{∗} such that η(−1) = (−1)^{g+1}. Then the homomorphism

G_{1} → (G_{1} × G_{2})^{0}/K, σ_{1} ↦ (σ_{1}, η(σ_{1})I_{2}) mod K

factors through a homomorphism P → (G_{1} × G_{2})^{0}/K and defines a splitting of the extension (25). Since elements of the form (σ_{1}, η(σ_{1})I_{2}) commute with elements of N, we see that the extension is trivial when it splits. It is easy to see that the converse is also true. Since the trivial η works when g is odd, we obtain that the extension always splits in this case. Assume g is even and G_{1} admits a 1-dimensional representation η with η(−I_{2}) = −1. If its kernel is trivial, G_{1} is isomorphic to a subgroup of ℂ^{∗}, hence cyclic. Otherwise, the kernel is a subgroup of SL(2) not containing the center. It must be a cyclic subgroup of odd order. The image is a cyclic group. Thus G_{1} is either cyclic, or a binary dihedral group D_{2n} with n odd.

To summarize we have proved the following.

Proposition 5.3. The group of automorphisms of an exceptional conic bundle (24) is isomorphic to an extension N•P, where P is the subgroup of PGL(2) leaving the set of zeroes of F_{2g+2}(T_{0}, T_{1}) invariant and N ≅ ℂ^{∗} : 2 is a group of matrices with determinant ±1 leaving T_{2}T_{3} invariant. Moreover, the extension splits and defines an isomorphism

Aut(Y_{g}) ≅ N × P

if and only if g is odd, or g is even and P is either a cyclic group or a dihedral group D_{4k+2}.

Now let G be a finite minimal subgroup of Aut(Y_{g}). Assume first that Aut(Y_{g}) ≅ N ×P. Let N′ be the projection of G to N and P′ be the projection to P. Since G is minimal, N′ contains an element which switches V (T_{2}) and V (T_{3}). Thus N′ is isomorphic to a dihedral group D_{2n}. Applying Goursat’s Lemma we obtain that

G ≅ N′ △_{D} Q,

where D is a common quotient of N′ and Q. If N′ is a dihedral group, then D is either dihedral group or a cyclic of order 2. Using Goursat’s Lemma it is easy to list all possible subgroups. We leave it as an exercise to the reader.

If Aut(Y_{g}) is not isomorphic to the direct product N × P, we can only say that

G ≅ H•Q,

where H is a subgroup of D_{2n} or Q_{4n}, and Q is a polyhedral group. Note that we can write these extensions in the form n•(2•Q) or n•(2^{2}•Q).

Example 5.4. Let π : S → ℙ^{1} be an exceptional conic bundle with g = 1. It has 4 singular fibres. According to the first construction, the blow up S′ of S at the four singular points of the singular fibres admits an elliptic fibration f : S′ → ℙ^{1} with two singular fibres of type I_{0}^{∗} in Kodaira’s notation. The j-invariant of the fibration is zero, and after the degree 2 base change ℙ^{1} → ℙ^{1} ramified at 2 points, the surface becomes isomorphic to the product E × ℙ^{1}, where E is an elliptic curve. Its j-invariant corresponds to the cross ratio of the 4 points, where a section of π with self-intersection −2 intersects the singular fibres. Conversely, starting from the product, we can divide it by an elliptic involution, to get the conic bundle. This is our second construction.

According to the third construction, the surface can be given by an equation

F_{4}(T_{0}, T_{1}) + T_{2}T_{3} = 0

in the weighted projective space ℙ(1, 1, 2, 2). The projection to (T_{0}, T_{1}) is a rational map undefined at the four points P_{i} = (a, b, 0, 0), where F_{4}(a, b) = 0. After we blow them up we get the conic bundle. The projection to (T_{2}, T_{3} is a rational map undefined at the two singular points (0, 0, 1, 0) and (0, 0, 0, 1). After we blow them up, we get the elliptic fibration. We have two obvious commuting involutions σ_{1} = [t_{0}, t_{1}, −t_{3}, −t_{2}] and σ_{2} = [t_{0}, t_{1}, t_{3}, t_{2}]. The locus of fixed points of each of them is an elliptic curve with equation T_{2} = 0 or T_{3} = 0. The group ⟨σ_{1}, σ_{2}⟩ ≅ 2^{2} permutes these two curves.

The groups of automorphisms of S is easy to describe. It follows from Proposition 5.3 that G is a finite subgroup of P ×K, where P is a subgroup of PGL(2) leaving the zeros of F_{4} invariant and K is a subgroup of GL(2) leaving the zeroes of t_{2}t_{3} invariant. First we choose coordinates T_{0}, T_{1} to write F_{4} in the form T_{0}^{4} + T_{1}^{4} + aT_{0}^{2}T_{1}^{2}, a^{2} ≠ 4. It is always possible if F_{4} has 4 distinct roots (true in our case). Let P be the subgroup of PGL(2) leaving the set of zeroes of F_{4} invariant. It is one of the following groups 1, 2, 4, 2^{2}, D_{8}, A_{4}. If a^{2} ≠ 0, −12, 36, then P is a subgroup of 2^{2}. If a^{2} = 0, 36, then P is a subgroup of D_{8}. If a^{2} = −12, then P is a subgroup of A_{4}.

Suppose a is not exceptional. Let P̄ be the corresponding binary group. Then it leaves F_{4} invariant, so G is a subgroup of K′ × P, where K′ consists of matrices leaving t_{2}t_{3} invariant. It is generated by diagonal matrices with determinant 1 and the transformations [t_{1}, t_{0}]. Thus

G ⊂ D_{2n} × 2^{2},

where we use that a a finite subgroup of K′ is either cyclic or binary dihedral.

Suppose a^{2} = 0, 36 and G contains an element of order 4. The form can be transformed to the form T_{0}^{4} + T_{1}^{4}. The value of the character at an element τ of order 4 is equal to −1. We obtain

G ⊂ \frac{1}{2}[Q × P] ≅ (Q′ × P′)^{•}2 ≅ Q′_{•}P

where Q is a finite subgroup of K and the diagonal product is taken with respect to the subgroup Q′ = Q ∩ K′ of Q and the subgroup P ∩ 2^{2} of index 2 of P. The group Q′ is cyclic or dihedral.

Suppose a = −2\sqrt{3}i and G contains an element g of order 3 given by [\frac{1−i}{2}t_{0} + \frac{1−i}{2}t_{1}, −\frac{1+i}{2}t_{0} − \frac{1+i}{2}t_{1}]. Its character is defined by χ(g) = ϵ_{3}. We obtain

G ⊂ \frac{1}{3}[Q × P],

where the diagonal product is taken with respect to the subgroup Q = χ_{2}^{−1}(μ_{3}) of K and the subgroup P′ = P ∩ 2^{2} of index 3 of P. Again G is a subgroup of D_{2n}•P.

5.4 Minimal conic bundles G-surfaces

Now assume (S, G) is a minimal G-surface such that S admits a conic bundle map ϕ : S → ℙ^{1}. As we had noticed before the number of singular fibres k is greater or equal to 4. Thus

K_{S}^{2} = 8 − k ≤ 4.  (26)

Let (S, G) be a rational G-surface and

a : G → O(Pic(X)), g ↦ (g^{∗})^{−1}  (27)

be the natural representation of G in the orthogonal group of the Picard group. We denote by G_{0} the kernel of this representation. Since k > 2 and G_{0} fixes any component of a singular fibre, it acts identically on the base of the conic bundle fibration. Since G_{0} fixes the divisor class of a section, and sections with negative self-intersection do not move in a linear system, we see that G_{0} fixes pointwisely any section with negative self-intersection. If we consider a section as a point of degree 1 on the generic fibre, we see that G_{0} must be is a cyclic group.

Proposition 5.5. Assume G_{0} ≠ {1}. Then S is an exceptional conic bundle.

Proof. Let g_{0} be a non-trivial element from G_{0}. Let E be a section with E^{2} = −n < 0. Take an element g ∈ G such that E′ = g(E) ≠ E. Since g_{0} has two fixed points on each component of a singular fibre we obtain that E and E′ do not intersect the same component. By Lemma 5.1, we obtain that k = 2n. Now we blow down n components in n fibres intersecting E and n components in the remaining n fibres intersecting E′ to get a minimal ruled surface with two disjoint sections with self-intersection 0. It must be isomorphic to F_{0}. So, we see that S is an exceptional conic bundle (Construction 1) with n = g + 1.

From now on in this section, we assume that G_{0} = {1}.

Let S_{η} be the general fibre of ϕ. By Tsen’s theorem it is isomorphic to ℙ_{K}^{1}, where K = ℂ(t) is the field of rational functions of the base. Consider S_{η} as a scheme over ℂ. Then

Aut_{ℂ}(S_{η}) ≅ Aut_{K}(S_{η}) : PGL(2) ≅ dJ(2),

where dJ(2) is a de Jonquiéres subgroup of Cr(2) and Aut_{K}(S_{η}) ≅ PGL(2, K). A finite minimal group G of automorphisms of a conic bundle is isomorphic to a subgroup of Aut_{ℂ}(S_{η}). Let G_{K} = G ∩ Aut_{K}(S_{η}) and G_{B} ≅ G/G_{K} be the image of G in PGL(2). We have an extension of groups

1 → G_{K} → G → G_{B} → 1  (28)

Let R be the subgroup of Pic(S) spanned by the divisor classes of R_{i} − R′_{i}, i = 1, …, k. It is obviously G-invariant and R_{ℚ} is equal to the orthogonal complement of Pic(S)_{ℚ}^{G} in Pic(S)_{ℚ}. The orthogonal group of the quadratic lattice R is isomorphic to the wreath product 2 ≀ S_{k}. The normal subgroup 2^{k} consists of transformations which switch some of the R_{i}’s with R′_{i}. A subgroup isomorphic to S_{k} permutes the classes R_{i} − R′_{i}.

Lemma 5.6. Let G be a minimal group of automorphisms of S. There exists an element g ∈ G_{K} of order 2 which switches the components of some singular fibre.

Proof. Since G is minimal, the G-orbit of any R_{i} cannot consist of disjoint components of fibres. Thus it contains a pair R_{j}, R′_{j} and hence there exists an element g ∈ G such that g(R_{j}) = R′_{j}. If g is of odd order 2k + 1, then g^{2k} and g^{2k+1} fix R_{j}, hence g fixes R_{j}. This contradiction shows that g is of even order 2m. Replacing g by an odd power, we may assume that g is of order m = 2^{a}.

Assume a = 1. Obviously the singular point p = R_{j} ∩ R′_{j} of the fibre belongs to the fixed locus S^{g} of g. Suppose p is an isolated fixed point. Then we can choose local coordinates at p such that g acts by (z_{1}, z_{2}) ↦ (−z_{1}, −z_{2}), and hence acts identically on the tangent directions. So it cannot switch the components. Thus S^{g} contains a curve not contained in fibres which passes through p. This implies that g ∈ G_{K}.

Suppose a > 1. Replacing g by g′ = g^{m/2} we get an automorphism of order 2 which fixes the point x_{j} and the components R_{j}, R′_{j}. Suppose S^{g′} contains one of the components, say R_{j}. Take a general point y ∈ R_{j}. We have g′(g(y)) = g(g′(y)) = g(y). This shows that g′ fixes R′_{j} pointwisely. Since S^{g′} is smooth, this is impossible. Thus g′ has 3 fixed points y, y′, p on F_{j}, two on each component. Suppose y is an isolated fixed point lying on R_{j}. Let π : S → S′ be the blowing down of R_{j}. The element g′ descends to an automorphism of order 2 of S′ which has an isolated fixed point at q = π(R_{j}). Then it acts identically on the tangent directions at q, hence on R_{j}. This contradiction shows that S^{g′} contains a curve intersecting F_{j} at y or at p, and hence g′ ∈ G_{K}. Since g′ is an even power, it cannot switch any components of fibres. Hence g′ acts identically on R and hence on all Pic(S). Thus g′ = g^{m/2} = 1 (recall that we assume that G_{0} is trivial), contradicting the definition of order of g.

The restriction of the homomorphism G → O(R) ≅ 2^{k} : S_{k} to G_{K} defines a surjective homomorphism

ρ : G_{K} → 2^{s}, s ≤ k.

An element from Ker(ρ) acts identically on R and hence on Pic(S). By Lemma 5.6, G_{K} is not trivial and s > 0. A finite subgroup of PGL(2, K) does not admit a surjective homomorphism to 2^{s} for s > 2. Thus s = 1 or 2.

Case 1: s = 1.

Let Σ′ be the non-empty subset of Σ such that G_{K} switches the components of fibres over Σ′. Since G_{K} is a normal subgroup of G, the set Σ′ is a G-invariant set. If Σ ≠ Σ′, we repeat the proof of Lemma 5.6 starting from some component R_{i} of a fibre over a point x ∉ Σ′, and find an element in G_{K} of even order which switches the components of maybe another fibre F_{x}, where x ∉ Σ′. Since G_{K} = 2, we get a contradiction.

Let G_{K} = ⟨h⟩. The element h fixes two points on each nonsingular fibre. The closure of these points is a one-dimensional component C of S^{h}. It is a smooth bi-section of the fibration. Since we know that h switches all components, its trace on the subgroup R generated by the divisor classes R_{i} − R′_{i} is equal to −k. Thus its trace on H^{2}(S, ℚ) is equal to 2 − k. Applying the Lefschetz fixed-point-formula, we get e(S^{h}) = 4 − k. If C is the disjoint union of two components, then S^{h} consists of k isolated fixed points (the singular points of fibres) and C. We get e(S^{h}) = 4 + k. This contradiction shows that C is irreducible and e(C) = 4 − k. Since h fixes pointwisely C and switches the components R_{i} and R′_{i}, the intersection point R_{i} ∩ R′_{i} must be on C. Thus the projection C → ℙ^{1} has ≥ k ramification points. Hence 4 − k = e(C) = 4 − (2 + 2g(C)) ≤ 4 − k. This shows that k = 2g(C) + 2, i.e. the singular points of fibres are the ramification points of the g_{2}^{1}.

Case 2: s = 2.

Let g_{1}, g_{2} be two elements from G_{K} which are mapped to generators of the image of G_{K} in 2^{k}. Let C_{1} and C_{2} be one-dimensional components of the sets S^{g_{1}} and S^{g_{2}}. As in the previous case we show that C_{1} and C_{2} are smooth hyperelliptic curves of genera g(C_{1}) and g(C_{2}). Let Σ_{1} and Σ_{2} be the sets of branch points of the corresponding double covers. As in the previous case we show that Σ = Σ_{1} ∪ Σ_{2}. For any point x ∈ Σ_{1} ∩ Σ_{2} the transformation g_{3} = g_{1}g_{2} fixes the components of the fibre F_{x}. For any point x ∈ Σ_{1} \ Σ_{2}, g_{3} switches the components of F_{x}. Let C_{3} be the one-dimensional component of S^{g_{3}} and Σ_{3} be the set of branch points of g_{3}. We see that Σ_{i} = Σ_{j} + Σ_{k} for distinct i, j, k, where Σ_{j} + Σ_{k} = (Σ_{j} ∪ Σ_{k}) \ (Σ_{j} ∩ Σ_{k}). This implies that there exist three binary forms p_{1}(t_{0}, t_{1}), p_{2}(t_{0}, t_{1}), p_{3}(t_{0}, t_{1}), no two of which have a common root, such that Σ_{1} = V (p_{2}p_{3}), Σ_{2} = V (p_{1}p_{3}), Σ_{3} = V (p_{1}p_{2}). Setting d_{i} = deg p_{i}, we get

2g(C_{i}) + 2 = d_{j} + d_{k}.

Let us summarize what we have learnt.

Theorem 5.7. Let G be a minimal finite group of automorphisms of a conic bundle f : S → ℙ^{1} with a set Σ of singular fibres. Assume G_{0} = {1}. Then k = #Σ > 2 and one of the following cases occurs.

(1) G = 2P, where the central involution h fixes pointwisely an irreducible smooth bisection C of π and switches the components in all fibres. The curve C is a curve of genus g = (k − 2)/2. The conic bundle projection defines a g_{2}^{1} on C with ramification points equal to singular points of fibres. The group P is isomorphic to the group of automorphisms of C modulo the involution defined by the g_{2}^{1}.

(2) G ≅ 2^{2}•P, each nontrivial element g_{i} of the subgroup 2^{2} fixes pointwisely an irreducible smooth bisection C_{i}. The set Σ is partitioned in 3 subsets Σ_{1}, Σ_{2}, Σ_{3} such that the projection f : C_{i} → ℙ^{1} ramifies over Σ_{j} + Σ_{k}, i ≠ j ≠ k. The group P is subgroup of Aut(ℙ^{1}) leaving the set Σ and its partition into 3 subsets Σ_{i} invariant.

It follows from Lemma 4.4 that in Case 1 the non-split extension is isomorphic to a binary polyhedral group, unless G = O or D_{2n}, where n is even.

Remark 5.8. It follows from the previous description that any abelian group G of automorphisms of a conic bundle must be a subgroup of some extension Q•P, where Q is a dihedral, binary dihedral or cyclic group, and P is a polyhedral group. This implies that G is either a cyclic group, or a group 2 × m, or 2^{2} × m, or 2^{4}. All these groups occur (see Example 5.12 and [6]).

5.5 Automorphisms of hyperelliptic curves

We consider a curve of genus g equipped with a linear series g_{2}^{1} as a curve C of degree 2g + 2 in ℙ(1, 1, g + 1) given by an equation

T_{2}^{2} + F_{2g+2}(T_{0}, T_{1}) = 0.

An automorphism σ of C is defined by a transformation

(t_{0}, t_{1}, t_{2}) ↦ (at_{1} + bt_{0}, ct_{1} + dt_{0}, αt_{2}),

where ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2) and F (aT_{0}+bT_{1}, cT_{0}+dT_{1}) = α^{2}F (T_{0}, T_{1}). So to find the group of automorphisms of C we need to know relative invariants Φ(T_{0}, T_{1}) for finite subgroups P̄ of SL(2, ℂ) (see [48]). The set of relative invariants is a finitely generated ℂ-algebra. Its generators are called Gründformen. We will list the Gründformen (see [48]). We will use them later for the description of automorphism groups of Del Pezzo surfaces of degree 1.

• P̄ is a cyclic group of order n.

A generator is given by the matrix

g = (\begin{matrix}ϵ_{n} & 0 \\ 0 & ϵ_{n}^{−1} \end{matrix}) .

The Gründformen are t_{0} and t_{1} with characters determined by

χ_{1}(g) = ϵ_{n}, χ_{2}(g) = ϵ_{n}^{−1}.

• P̄ ≅ Q_{4n} is a binary dihedral group of order 4n.

Its generators are given by the matrices

g_{1} = (\begin{matrix}ϵ_{2n} & 0 \\ 0 & ϵ_{2n}^{−1} \end{matrix}) , g_{2} = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) .

The Gründformen are

Φ_{1} = t_{0}^{n} + t_{1}^{n}, Φ_{2} = t_{0}^{n} − t_{1}^{n}, Φ_{3} = t_{0}t_{1}.  (29)

The generators g_{1} and g_{2} act on the Gründformen with characters

χ_{1}(g_{1}) = χ_{2}(g_{1}) = −1, χ_{1}(g_{2}) = χ_{2}(g_{2}) = i^{n},
χ_{3}(g_{1}) = 1, χ_{3}(g_{2}) = −1.

• P̄ is a binary tetrahedral group of order 24.

Its generators are given by the matrices

g_{1} = (\begin{matrix}ϵ_{4} & 0 \\ 0 & ϵ_{4}^{−1} \end{matrix}) , g_{2} = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) , g_{3} = \frac{1}{i − 1} (\begin{matrix}i & i \\ 1 & −1 \end{matrix}) .

The Gründformen are

Φ_{1} = t_{0}t_{1}(t_{0}^{4} − t_{1}^{4}), Φ_{2}, Φ_{3} = t_{0}^{4} ± 2\sqrt{−3}t_{0}^{2}t_{1}^{2} + t_{1}^{4}.

The generators g_{1}, g_{2}, g_{3} act on the Gründformen with characters

χ_{1}(g_{1}) = χ_{1}(g_{2}) = χ_{1}(g_{3}) = 1,

χ_{2}(g_{1}) = χ_{2}(g_{2}) = 1, χ_{2}(g_{3}) = ϵ_{3},
χ_{3}(g_{1}) = χ_{3}(g_{2}) = 1, χ_{3}(g_{3}) = ϵ_{3}^{2}.

• P̄ is a binary octahedral group of order 48.

Its generators are

g_{1} = (\begin{matrix}ϵ_{8} & 0 \\ 0 & ϵ_{8}^{−1} \end{matrix}) , g_{2} = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) , g_{3} = \frac{1}{i − 1} (\begin{matrix}i & i \\ 1 & −1 \end{matrix}) .

The Gründformen are Φ_{1} = t_{0}t_{1}(t_{0}^{4} − t_{1}^{4}), Φ_{2} = t_{0}^{8} + 14t_{0}^{4}t_{1}^{4} + t_{1}^{8}, Φ_{3} = (t_{0}^{4} + t_{1}^{4})((t_{0}^{4} + t_{1}^{4})^{2} − 36t_{0}^{4}t_{1}^{4}). The generators g_{1}, g_{2}, g_{3} act on the Gründformen with characters

χ_{1}(g_{1}) = −1, χ_{1}(g_{2}) = χ_{1}(g_{3}) = 1,

χ_{2}(g_{1}) = χ_{2}(g_{2}) = χ_{2}(g_{3}) = 1,
χ_{3}(g_{1}) = −1, χ_{3}(g_{2}) = χ(g_{3}) = 1.

• P̄ is a binary icosahedral group of order 120.

Its generators are

g_{1} = (\begin{matrix}ϵ_{10} & 0 \\ 0 & ϵ_{10}^{−1} \end{matrix}) , g_{2} = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) , g_{3} = \frac{1}{\sqrt{5}} (\begin{matrix}ϵ_{5} − ϵ_{5}^{4} & ϵ_{5}^{2} − ϵ_{5}^{3} \\ ϵ_{5}^{2} − ϵ_{5}^{3} & −ϵ_{5} + ϵ_{5}^{4} \end{matrix}) .

The Gründformen are

Φ_{1} = t_{0}^{30} + t_{1}^{30} + 522(t_{0}^{25}t_{1}^{5} − t_{0}^{5}t_{1}^{25}) − 10005(t_{0}^{20}t_{1}^{10} + t_{0}^{10}t_{1}^{20}),

Φ_{2} = −(t_{0}^{20} + t_{1}^{20}) + 228(t_{0}^{15}t_{1}^{5} − t_{0}^{5}t_{1}^{15}) − 494t_{0}^{10}t_{1}^{10},
Φ_{3} = t_{0}t_{1}(t_{0}^{10} + 11t_{0}^{5}t_{1}^{5} − t_{1}^{10}).

Since P/(±1) ≅ A_{5} is a simple group and all Gründformen are of even degree, we easily see that the characters are trivial.

5.6 Commuting de Jonquiéres involutions

Recall that a de Jonquiéres involution IH_{g+2} is regularized by an automorphism of the surface S which is obtained from F_{1} by blowing up 2g +2 points. Their images on ℙ^{2} are the 2g + 2 fixed points of the involution of H_{g+2}. Let π : S → X = S/IH_{g+2}. Since the fixed locus of the involution is a smooth hyperelliptic curve of genus g, the quotient surface X is a nonsingular surface. Since the components of singular fibres of the conic bundle on S are switched by IH_{g+2}, their images on X are isomorphic to ℙ^{1}. Thus X is a minimal ruled surface F_{e}. What is e?

Let C̄ = π(C) and Ē = π(E), where E is the exceptional section on S. The curve Ē is a section on X whose preimage in the cover splits. It is either tangent to C̄ at any of its of g intersection points (since IH_{g+2}(E) · E = g) or is disjoint from C̄ if g = 0. Let s be the divisor class of a section on F_{e} with self-intersection −e and f be the class of a fibre. It is easy to see that

C̄ = (g + 1 + e)f + 2s, Ē = \frac{g + e − 1}{2}f + s.

Let R̄ be a section with the divisor class s. Suppose R̄ = Ē, then R̄ · C̄ = g + 1 − e = 2g implies g = 1 − e, so (g, e) = (1, 0) or (0, 1). In the first case, we get an elliptic curve on F_{0} with divisor class 2f + 2s and S is nonexceptional conic bundle with k = 4. In the second case S is the conic bundle (non-minimal) with k = 2.

Assume that (g, e) ≠ (1, 0). Let R = π^{−1}(R̄) be the pre-image of R̄. We have R^{2} = −2e. If it splits into two sections R_{1} + R_{2}, then R_{1} · R_{2} = C̄ · R̄ = g +1−e, hence −2e = 2(g +1−e)+2R_{1}^{2} gives R_{1}^{2} = −g −1. Applying Lemma (5.1), we get R_{1}·R_{2} = g+1−e = g−1+(2g+2−a)/2 = −a/2, where a ≥ 0. This gives e = g +1, but intersecting Ē with R̄ we get e ≤ g −1. This contradiction shows that R̄ does not split, and hence R is an irreducible bisection of the conic bundle with R^{2} = −2e. We have R·E = (g−e−1)/2, R·R_{i} = R·R′_{i} = 1, where R_{i} + R′_{i} are reducible fibres of the conic fibration.

This shows that the image of R in the plane is a hyperelliptic curve H′_{g′+2} of degree d = (g − e + 3)/2 and genus g′ = d − 2 = (g − e − 1)/2 with the point q of multiplicity g′. It also passes through the points p_{1}, …, p_{2g+2}. Its Weierstrass points p′_{1}, …, p′_{2g′+2} lie on H_{g+2}. Here we use the notation from 2.3. Also the curve H′_{g′+2} is invariant with respect to the de Jonquiéres involution.

Write the equation of H′_{g′+2} in the form

A_{g′}(T_{0}, T_{1})T_{2}^{2} + 2A_{g′+1}(T_{0}, T_{1})T_{2} + A_{g′+2}(T_{0}, T_{1}) = 0.  (30)

It follows from the geometric definition of the de Jonquiéres involution that we have the following relation between the equations of H′_{g′+2} and H_{g+2} (cf. [17], p.126)

F_{g}A_{g′+2} − 2F_{g+1}A_{g′+1} + F_{g+2}A_{g′} = 0.  (31)

Consider this as a system of linear equations with coefficients of A_{g′+2}, A_{g′+1}, A_{g′} considered as the unknowns. The number of the unknowns is equal to (3g − 3e + 9)/2. The number of the equations is (3g − e + 5)/2. So, for a general H_{g+2} we can solve these equations only if g = 2k + 1, e = 0, d = k + 2 or g = 2k, e = 1, d = k +1. Moreover, in the first case we get a pencil of curves R satisfying these properties, and in the second case we have a unique such curve (as expected). Also the first case covers our exceptional case (g, e) = (1, 0).

For example, if we take g = 2 we obtain that the six Weierstrass points p_{1}, …, p_{6} of H_{g+2} must be on a conic. Or, if g = 3, the eight Weierstrass points together with the point q must be the base points of a pencil of cubics. All these properties are of course not expected for a general set of 6 or 8 points in the plane.

To sum up, we have proved the following.

Theorem 5.9. Let H_{g+2} be a hyperelliptic curve of degree g + 2 and genus g defining a de Jonquiéres involution IH_{g+2}. View this involution as an automorphism τ of order 2 of the surface S obtained by blowing up the singular point q of H_{g+2} and its 2g + 2 Weierstrass points p_{1}, …, p_{2g+2}. Then

(i) the quotient surface X = S/(τ) is isomorphic to F_{e} and the ramification curve is C = S^{τ};

(ii)if H_{g+2} is a general hyperelliptic curve then e = 0 if g is odd and e = 1 if

g is even; (iii)the branch curve C̄ of the double cover S → F_{e} is a curve from the divisor

class (g + 1 + e)f + 2s; (iv)there exists a section from the divisor class \frac{g+e−1}{2}f + s which is tangent

to C̄ at each g intersection points unless g = 0, e = 1 in which case it is disjoint from C̄;

(v) the reducible fibres of the conic bundle on S are the pre-images of the 2g+2 fibres from the pencil |f| which are tangent to C̄;

(vi)the pre-image of a section from the divisor class s either splits if (g, e) = (1, 0) or a curve of genus g = 0, or a hyperelliptic curve C′ of genus g′ = (g − e − 1)/2 ≥ 1 which is invariant with respect to τ. It intersects the hyperelliptic curve C at its 2g′ + 2 Weierstrass points.

(vii)the curve C′ is uniquely defined if e > 0 and varies in a pencil if e = 0.

Let IH′_{g′+2} be the de Jonquiéres involution defined by the curve H′_{g′+2} from equation (30). Then it can be given in affine coordinates by formulas (7), where F_{i} is replaced with A_{i}. Thus we have two involutions defined by the formulas

IH_{g+2} : (x′, y′) = (x, \frac{−yP_{g+1}(x) − P_{g+2}(x)}{P_{g}(x)y + P_{g+1}(x)}),  (32)
IH′_{g′+2} : (x′, y′) = (x, \frac{−yQ_{g′+1}(x) − Q_{g′+2}(x)}{Q_{g}(x)y + Q_{g′+1}(x)}),

where P_{i} are the dehomogenizations of the F_{i}’s and Q_{i} are the dehomogenizations of the A_{i}’s. Composing them in both ways we see that the relation (31) is satisfied if and only if the two involutions commute. Thus a de Jonquiéres involution can be always included in a group of de Jonquiéres transformations isomorphic to 2^{2}. In fact, for a general IH_{g+2} there exists a unique such group if g is even and there is a ∞^{1} such groups when g is odd. It is easy to check that the involution IH_{g+2} ∘ H′_{g′+2} is the de Jonquiéres involution defined by the third hyperelliptic curve with equation

det (\begin{matrix}F_{g} & F_{g+1} & F_{g+2} \\ A_{g′} & A_{g′+1} & A_{g′′+2} \\ 1 & −T_{2} & T_{2}^{2} \end{matrix}) = B_{g′′}T_{2}^{2} + 2B_{g′′−1}T_{2} + B_{g′′+2} = 0, (33)

(cf. [17], p.126).

If we blow up the Weierstrass point of the curve C′ (the proper transform of H′_{g′+2} in S), then we get a conic bundle surface S′ from case (2) of Theorem 5.7.

5.7 A question on extensions

It still remains to decide which extensions

1 → G_{K} → G → G_{B} → 1  (34)

describe minimal groups of automorphisms of conic bundles. We do not have the full answer and only make a few remarks and examples. Lemma 4.4 helps to decide on splitting in the case when K is abelian and central.

Example 5.10. Consider a de Jonquiéres transformation

dj_{P} : (x, y) ↦ (x, P (x)/y),

where P (T_{1}/T_{0}) = T_{0}^{−2g}F_{2g+2}(T_{0}, T_{1}) is a dehomogenization of ahomogeneous polynomial F_{2g+2}(T_{0}, T_{1}) of degree 2g + 2 defining a hyperelliptic curve of genus g. Choose F_{2g+2} to be a relative invariant of a binary polyhedral group P̄ with character χ : P̄ → ℂ^{∗}. We assume that χ = α^{2} for some character α : P̄ → ℂ^{∗}. For any g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ P̄ define the transformation

g : (x, y) ↦ (\frac{ax + b}{cx + d}, α(g)(cx + d)^{−g−1}y).

We have

P (\frac{ax + b}{cx + d}) = α^{2}(g)(cx + d)^{−2g−2}P (x).

It is immediate to check that g and dj_{p} commute. The matrix −I_{2} defines the transformation g_{0} : (x, y) ↦ (x, α(−I_{2})(−1)^{g+1}y). So, if

α(−I_{2}) = (−1)^{g+1},

the action of P̄ factors through P and together with dj_{P} generate the group 2 × P. On the other hand, if α(−I_{2}) = (−1)^{g}, we get the group G = 2 × P̄. In this case the group G is regularized on an exceptional conic bundle with G_{0} ≅ 2. The generator corresponds to the transformation g_{0}.

Our first general observation is that the extension G = 2P always splits if g is even, and of course, if P is a cyclic group of odd order. In fact, suppose G does not split. We can always find an element g ∈ G which is mapped to an element ḡ in P of even order 2d such that g^{2d} = g_{0} ∈ G_{K}. Now g_{1} = g^{d} defines an automorphism of order 2 of the hyperelliptic curve C = S^{g_{0}} with fixed points lying over two fixed points of ḡ in ℙ^{1}. None of these points belong to Σ, since otherwise g_{0}, being a square of g_{1}, cannot switch the components of the corresponding fibre. Since g_{1} has two fixed points on the invariant fibre and both of them must lie on C, we see that g_{1} has 4 fixed points. However this contradicts the Hurwitz formula.

Recall that a double cover f : X → Y of nonsingular varieties with branch divisor W ⊂ Y is given by an invertible sheaf L together with a section s_{W} ∈ Γ (Y, L^{2}) with zero divisor W. Suppose a group G acts on Y leaving invariant W. A lift of G is a group G̃ of automorphisms of X such that it commutes with the covering involution τ of X and the corresponding homomorphism G̃ → Aut(Y) is an isomorphism onto the group G.

The following lemma is well-known and is left to the reader.

Lemma 5.11. A subgroup G ⊂ Aut(Y) admits a lift if and only if L admits a G-linearization and in the corresponding representation of G in Γ (Y, L^{2}) the section s_{W} is G-invariant.

Example 5.12. Let p_{i}(t_{0}, t_{1}), i = 0, 1, 2, be binary forms of degree d. Consider a curve C in F_{0} ≅ ℙ^{1} × ℙ^{1} given by an equation

F = p_{0}(t_{0}, t_{1})x_{0}^{2} + 2p_{1}(t_{0}, t_{1})x_{0}x_{1} + p_{2}(t_{0}, t_{1})x_{1}^{2} = 0.

Assume that the binary form D = p_{1}^{2} − p_{0}p_{2} does not have multiple roots. Then C is a nonsingular hyperelliptic curve of genus d − 1. Suppose d = 2a is even so that the genus of the curve is odd. Let P be a polyhedral group not isomorphic to a cyclic group of odd order. Let V = Γ (ℙ^{1}, O_{ℙ^{1}}) and ρ : P → GL(S^{2a}V ⊗S^{2}V) be its natural representation, the tensor product of the two natural representations of P in the space of binary forms of even degree. Suppose that F ∈ S^{2a}V ⊗ S^{2}V is an invariant. Consider the double cover S → F_{0} defined by the section F and the invertible sheaf L = O_{F_{0}}(a, 1). Now assume additionally that P does not have a linear representation in S^{a}V ⊗ V whose tensor square is equal to ρ. Thus L does not admit a P -linearization and we cannot lift P to a group of automorphisms of the double cover. However, the binary polyhedral group P̄ lifts. Its central involution acts identically on 𝔽_{0}, hence lifts to the covering involution of S. It follows from the discussion in the previous subsection that S is a non-exceptional conic bundle, and the group P̄ is a minimal group of automorphisms of S with G_{K} ≅ 2 and G_{B} ≅ P.

Here is a concrete example. Take

p_{0} = t_{0}t_{1}(t_{0}^{2} + t_{1}^{2}), p_{1} = t_{0}^{4} + t_{1}^{4}, p_{2} = t_{0}t_{1}(t_{0}^{2} − t_{1}^{2}).

Let P̄ ⊂ SL(2) be a cyclic group of order 4 that acts on the variables t_{0}, t_{1} via the transformation [it_{0}, −it_{1}] and on the variables x_{0}, x_{1} via the transformation [ix_{0}, −ix_{1}]. Then P̄ acts on S^{2}V ⊗ V via [−1, 1, −1] ⊗ [i, −i]. The matrix −I_{2} acts as 1 ⊗ −1 and hence P = P̄/(±I_{2}) does not act on S^{2}V ⊗ V. This realizes the cyclic group C_{4} as a minimal group of automorphisms of a conic bundle with k = 2g + 2 = 8.

The previous example shows that for any g ≡ 1 mod 4 one can realize a binary polyhedral group P̄ = 2.P as a minimal group of automorphisms of a conic bundle with 2g + 2 singular fibres. We do not know whether the same is true for g ≡ 3 mod 4.

Example 5.13. Let p_{i}(t_{0}, t_{1}), i = 1, 2, 3, be three binary forms of even degree d with no multiple roots. Assume no two have common zeroes. Consider a surface S in ℙ^{1} × ℙ^{2} given by a bihomogeneous form of degree (d, 2)

p_{1}(t_{0}, t_{1})z_{0}^{2} + p_{2}(t_{0}, t_{1})z_{1}^{2} + p_{3}(t_{0}, t_{1})z_{2}^{2} = 0,  (35)

The surface is nonsingular. The projection to ℙ^{1} defines a conic bundle structure on S with singular fibres over the zeroes of the polynomials p_{i}. The curves C_{i} equal to the pre-images of the lines z_{i} = 0 under the second projection are hyperelliptic curves of genus g = d − 1. The automorphisms σ_{1}, σ_{2} defined by the negation of one of the first two coordinates z_{0}, z_{1}, z_{2} form a subgroup of Aut(S) isomorphic to 2^{2}. Let P be a finite subgroup of SL(2, ℂ) and g ↦ g^{∗} be its natural action on the space of binary forms. Assume that p_{1}, p_{2}, p_{3} are relative invariants of P with characters χ_{1}, χ_{2}, χ_{3} such that we can write them in the form η_{i}^{2} for some characters η_{1}, η_{2}, η_{3} of P. Then P acts on S by the formula

g((t_{0}, t_{1}), (z_{0}, z_{1}, z_{2})) = ((g^{∗}(t_{0}), g^{∗}(t_{1})), (η_{1}(g)^{−1}z_{0}, η_{2}(g)^{−1}z_{1}, η_{3}(g)^{−1}z_{2})).

For example, let P = ⟨g⟩ be a cyclic group of order 4. We take p_{1} = t_{0}^{2} + t_{1}^{2}, p_{2} = t_{0}^{2} − t_{1}^{2}, p_{3} = t_{0}t_{1}. It acts on S by the formula

g : ((t_{0}, t_{1}), (z_{0}, z_{1}, z_{2})) ↦ ((it_{1}, it_{0}), (iz_{0}, z_{1}, iz_{2})).

Thus g^{2} acts identically on t_{0}, t_{1}, z_{1} and multiplies z_{0}, z_{2} by −1. We see that G_{K} = ⟨g^{2}⟩ and the extension 1 → G_{K} → G → G_{B} → 1 does not split. If we add to the group the transformation (t_{0}, t_{1}, z_{0}, z_{1}, z_{2}) ↦ (t_{0}, t_{1}, z_{0}, −z_{1}, z_{2}) we get a non-split extension 2^{2+1}.

On the other hand, let us replace p_{2} with t_{0}^{2} + t_{1}^{2} + t_{0}t_{1}. Define g_{1} as acting only on t_{0}, t_{1} by [it_{1}, it_{0}], g_{2} acts only on z_{0} by z_{0} ↦ −z_{0} and g_{3} acts only on z_{1} by z_{1} ↦ −z_{1}. We get the groups ⟨g_{1}, g_{2}⟩ = 2^{2} and ⟨g_{1}, g_{2}, g_{3}⟩ = 2^{3}.

In another example we take P to be the dihedral group D_{8}. We take p_{1} = t_{0}^{2} + t_{1}^{2}, p_{2} = t_{0}^{2} − t_{1}^{2}, p_{3} = t_{0}t_{1}. It acts on S by the formula

( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) : ((t_{0}, t_{1}), (z_{0}, z_{1}, z_{2})) ↦ ((it_{0}, −it_{1}), (iz_{0}, iz_{1}, z_{2})),

( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) : ((t_{0}, t_{1}), (z_{0}, z_{1}, z_{2})) ↦ ((it_{1}, it_{0}), (z_{0}, iz_{1}, z_{2})),

The scalar matrix c = −I_{2} belongs to G_{K} ≅ 2^{2} and the quotient P/(c) ≅ 2^{2} acts faithfully on the base. This gives a non-split extension 2^{2+2}.

Finally, let us take

p_{1} = t_{0}^{4} + t_{1}^{4}, p_{2} = t_{0}^{4} + t_{1}^{4} + t_{0}^{2}t_{1}^{2}, p_{3} = t_{0}^{4} + t_{1}^{4} − t_{0}^{2}t_{1}^{2}.

These are invariants for the group D_{4} acting via g_{1} : (t_{0}, t_{1}) ↦ (t_{0}, −t_{1}), g_{2} : (t_{0}, t_{1}) ↦ (t_{1}, t_{0}). Together with transformations σ_{1}, σ_{2} this generates the group 2^{4} (see another realization of this group in [8]).

6 Automorphisms of Del Pezzo surfaces

6.1 The Weyl group

Let S be a Del Pezzo surface of degree d not isomorphic to ℙ^{2} or F_{0}. It is isomorphic to the blow-up of N = 9 − d ≤ 8 points in ℙ^{2} satisfying the conditions of generality from section 3.4. The blow-up of one or 2 points is obviously non-minimal (since the exceptional curve in the first case and the proper transform of the line through the two points is G-invariant). So we may assume that S is a Del Pezzo surface of degree d ≤ 6.

Let π : S → ℙ^{2} be the blowing-up map. Consider the factorization (10) of π into a composition of blow-ups of N = 9 − d points. Because of the generality condition, we may assume that none of the points p_{1}, …, p_{N} is infinitely near, or, equivalently, all exceptional curves E_{i} are irreducible curves. We identify them with curves E_{i} = π^{−1}(p_{i}). The divisor classes e_{0} = [π^{∗}(line], e_{i} = [E_{i}], i = 1, …, N, form a basis of Pic(S). It is called a geometric basis.

Let

α_{1} = e_{0} − e_{1} − e_{2} − e_{3}, α_{2} = e_{1} − e_{2}, …, α_{N} = e_{N−1} − e_{N}.

For any i = 1, …, N define a reflection isometry s_{i} of the abelian group Pic(S)

s_{i} : x ↦ x + (x · α_{i})α_{i}.

Obviously, s_{i}^{2} = 1 and s_{i} acts identically on the orthogonal complement of α_{i}. Let W_{S} be the group of automorphisms of Pic(S) generated by the transformations s_{1}, …, s_{N}. It is called the Weyl group of S. Using the basis (e_{0}, …, e_{N}) we identify W_{S} with a group of isometries of the odd unimodular quadratic form q : ℤ^{N+1} → ℤ of signature (1, N) defined by

q_{N}(m_{0}, …, m_{N}) = m_{0}^{2} − m_{1}^{2} − ⋯ − m_{N}^{2}.

Since K_{S} = −3e_{0} + e_{1} + ⋯ + e_{N} is orthogonal to all α_{i}’s, the image of W_{S} in O(q_{N}) fixes the vector k_{N} = (−3, 1, …, 1). The subgroup of O(q_{N}) fixing k_{N} is denoted by W_{N} and is called the Weyl group of type E_{N}. The orthogonal complement R_{N} of k_{N} equipped with the restricted inner-product, is called the root lattice of W_{N}.

We denote by R_{S} the sublattice of Pic(S) equal to the orthogonal complement of the vector K_{S}. The vectors α_{1}, …, α_{N} form a ℤ-basis of R_{S}. By restriction the Weyl group W_{S} is isomorphic to a subgroup of O(R_{S}). A choice of a geometric basis α_{1}, …, α_{N} defines an isomorphism from R_{S} to the root lattice Q of a finite root system of type E_{N}(N = 6, 7, 8), D_{5}(N = 5), A_{4}(N = 4) and A_{2}+A_{1}(N = 3). The group W_{S} becomes isomorphic to the corresponding Weyl group W (E_{N}).

The next lemma is well-known and its proof goes back to Kantor [41] and Du Val [27]. We refer for modern proofs to [2] or [25].

Lemma 6.1. Let (e′_{0}, e′_{1}, …, e′_{N}) be another geometric basis in Pic(S) defined by a birational morphism π′ : S → ℙ^{2} and a choice of a factorization of π′ into a composition of blow-ups of points. Then the transition matrix is an element of W_{N}. Conversely, any element of W_{N} is a transition matrix of two geometric bases in Pic(S).

The next lemma is also well-known and is left to the reader.

Lemma 6.2. If d ≤ 5, then the natural homomorphism

ρ : Aut(S) → W_{S}

is injective.

We will use the known classification of conjugacy classes in the Weyl groups. According to [14] they are indexed by certain graphs. We call them Carter graphs. One writes each element w ∈ W as the product of two involutions w_{1}w_{2}, where each involution is the product of reflections with respect to orthogonal roots. Let R_{1}, R_{2} be the corresponding sets of such roots. Then the graph has vertices identified with elements of the set R_{1} ∪R_{2} and two vertices α, β are joined by an edge if and only if (α, β) ≠ 0. A Carter graph with no cycles is a Dynkin diagram. The subscript in the notation of a Carter graph indicates the number of vertices. It is also equal to the difference between the rank of the root lattice Q and the rank of its fixed sublattice Q^{(w)}.

Note that the same conjugacy classes may correspond to different graphs (e.g. D_{3} and A_{3}, or 2A_{3} + A_{1} and D_{4}(a_{1}) + 3A_{1}).

The Carter graph determines the characteristic polynomial of w. In particular, it gives the trace Tr_{2}(g) of g^{∗} on the cohomology space H^{2}(S, ℂ) ≅ Pic(S)⊗ℂ. The latter should be compared with the Euler-Poincarè characteristic of the fixed locus S^{g} of g by applying the Lefschetz fixed-point formula.

Graph  Order Characteristic polynomial
A_{k}  k + 1  t^{k} + t^{k−1} + ⋯ + 1
D_{k}  2k − 2  (t^{k−1} + 1)(t + 1)
D_{k}(a_{1}) l.c.m(2k − 4, 4)  (t^{k−2} + 1)(t^{2} + 1)
D_{k}(a_{2}) l.c.m(2k − 6, 6)  (t^{k−3} + 1)(t^{3} + 1)
⋮  ⋮  ⋮
D_{k}(a_{\frac{k}{2} −1})  even k  (t^{\frac{k}{2}} + 1)^{2}
E_{6}  12  (t^{4} − t^{2} + 1)(t^{2} + t + 1)
E_{6}(a_{1})  9  t^{6} + t^{3} + 1
E_{6}(a_{2})  6  (t^{2} − t + 1)^{2}(t^{2} + t + 1)
E_{7}  18  (t^{6} − t^{3} + 1)^{2}(t + 1)
E_{7}(a_{1})  14  t^{7} + 1
E_{7}(a_{2})  12  (t^{4} − t^{2} + 1)(t^{3} + 1)
E_{7}(a_{3})  30  (t^{5} + 1)(t^{2} − t + 1)
E_{7}(a_{4})  6  (t^{2} − t + 1)^{2}(t^{3} + 1)
E_{8}  30 t^{8} + t^{7} − t^{5} − t^{4} − t^{3} + t + 1
E_{8}(a_{1})  24  t^{8} − t^{4} + 1
E_{8}(a_{2})  20  t^{8} − t^{6} + t^{4} − t^{2} + 1
E_{8}(a_{3})  12  (t^{4} − t^{2} + 1)^{2}
E_{8}(a_{4})  18  (t^{6} − t^{3} + 1)(t^{2} − t + 1)
E_{8}(a_{5})  15 t^{8} − t^{7} + t^{5} − t^{4} + t^{3} − t + 1
E_{8}(a_{6}) 10 (t^{4} − t^{3} + t^{2} − t + 1)^{2}
E_{8}(a_{7})  12 (t^{4} − t^{2} + 1)(t^{2} − t + 1)^{2}
E_{8}(a_{8})  6  (t^{2} − t + 1)^{4}

Table 2. Carter graphs and characteristic polynomials

Tr_{2}(g) = s − 2 + ∑_{i∈I}(2 − 2g_{i}),  (36)

where S^{g} the disjoint union of smooth curves R_{i}, i ∈ I, of genus g_{i} and s isolated fixed points.

To determine whether a finite subgroup G of Aut(S) is minimal, we use the well-known formula from the character theory of finite groups

rank Pic(S)^{G} = \frac{1}{#G} ∑_{g∈G} Tr_{2}(g).

The tables for conjugacy classes of elements from the Weyl group W_{S} give the values of the trace on the lattice R_{S} = K_{S}^{⊥}. Thus the group is minimal if and only if the sum of the traces add up to 0.

6.2 Del Pezzo surfaces of degree 6

Let S be a Del Pezzo surface of degree 6. We fix a geometric basis e_{0}, e_{1}, e_{2}, e_{3} which is defined by a birational morphism π : S → ℙ^{2} with indeterminacy points p_{1} = (1, 0, 0), p_{2} = (0, 1, 0) and p_{3} = (0, 0, 1). The vectors

α_{1} = e_{0} − e_{1} − e_{2} − e_{3}, α_{2} = e_{1} − e_{2}, α_{3} = e_{2} − e_{3})

form a basis of the lattice R_{S} with Dynkin diagram of type A_{2} + A_{1}. The Weyl group

W_{S} = ⟨s_{1}⟩ × ⟨s_{2}, s_{3}⟩ ≅ 2 × S_{3}.

The representation ρ : Aut(S) → W_{S} is surjective. The reflection s_{1} is realized by the lift of the standard quadratic transformation τ_{1}. The reflection s_{2}( resp. s_{3}) is realized by the projective transformations [x_{1}, x_{0}, x_{2}] (resp. [x_{0}, x_{2}, x_{1}]). The kernel of ρ is the maximal torus T of PGL(3), the quotient of (ℂ^{∗})^{3} by the diagonal subgroup ℂ^{∗}. Thus

Aut(S) ≅ T : (S_{3} × 2) ≅ N(T) : 2,

where N(T) is the normalizer of T in PGL(2). It is easy to check that s_{1} acts on T as the inversion automorphism.

Let G be a minimal finite subgroup of Aut(S). Obviously, ρ(G) contains s_{1} and s_{2}s_{2} since otherwise G leaves invariant α_{1} or one of the vectors 2α_{1} + α_{2}, or α_{1} +2α_{2}. This shows that G∩N(T) is an imprimitive subgroup of PGL(3). This gives

Theorem 6.3. Let G be a minimal subgroup of a Del Pezzo surface of degree 6. Then

G = H_{•}⟨s_{1}⟩,

where H is an imprimitive finite subgroup of PGL(3).

Note that one the groups from the theorem is the group 2^{2} : S_{3} ≅ S_{4}. Its action on S given by the equation

x_{0}y_{0}z_{0} − x_{1}y_{1}z_{1} = 0

in (ℙ^{1})^{3} is given in [4].

6.3 Del Pezzo surfaces of degree d = 5.

In this case S is isomorphic to the blow-up of the reference points p_{1} = (1, 0, 0), p_{2} = (0, 1, 0), p_{3} = (0, 0, 1), p_{4} = (1, 1, 1). The lattice R_{S} is of type A_{4} and W_{S} ≅ S_{5}. It is known that the homomorphism ρ : Aut(S) → W_{S} is an isomorphism. We already know that it is injective. To see the surjectivity one can argue, for example, as follows.

Let τ be the standard quadratic transformation with base points p_{1}, p_{2}, p_{3}. It follows from its formula that the point p_{4} is a fixed point. We know that τ can be regularized on the Del Pezzo surface S′ of degree 6 obtained by the blow-up of the first three points. Since the pre-image of p_{4} in S′ is a fixed point, τ lifts to an automorphism of S. Now let ϕ be a projective transformation such that ϕ(p_{1}) = p_{1}, ϕ(p_{2}) = p_{2}, ϕ(p_{4}) = p_{3}. For example, we take A = [t_{0} − t_{2}, t_{1}, −t_{2}, −t_{2}]. Then the quadratic transformation ϕ^{−1}τϕ is not defined at the points p_{1}, p_{2}, p_{4} and fixes the point p_{3}. As above, it can be lifted to an involution of S. Proceeding in this way we define 4 involutions τ = τ_{1}, …, τ_{4} of S each fixes one of the exceptional curves. One checks that their images in the Weyl group W_{S} generate the group.

Another way to see the isomorphism Aut(S) ≅ S_{5} is to use a well-known isomorphism between S and the moduli space \overline{M_{0,5}} ≅ (ℙ^{1})^{5}//SL(2). The group S_{5} acts by permuting the factors.

Theorem 6.4. Let (S, G) be a minimal Del Pezzo surface of degree d = 5. Then G = S_{5}, A_{5}, 5 : 4, 5 : 2, or C_{5}.

Proof. As we have just shown Aut(S) ≅ W_{4} ≅ S_{5}. The group S_{5} acts on R_{S} ≅ ℤ^{4} by means of its standard irreducible 4-dimensional representation (view ℤ^{4} as a subgroup of ℤ^{5} of vectors with coordinates added up to zero and consider the representation of S_{5} by switching the coordinates). It is known that a maximal proper subgroup of S_{5} is equal (up to a conjugation) to one of three subgroups S_{4}, S_{3} × 2, A_{5}, 5 : 4. A maximal subgroup of A_{5} is either 5 × 2 or S_{3} or D_{10} = 5 : 2. It is easy to see that the groups S_{4} and S_{3} × 2 have invariant elements in the lattice Q_{4}. It is known that an element of order 5 in S_{5} is a cyclic permutation, and hence has no invariant vectors. Thus any subgroup G of S_{5} containing an element of order 5 defines a minimal surface (S, G). So, if (S, G) is minimal, G must be equal to one of the groups from the assertion of the theorem.

6.4 Automorphisms of a Del Pezzo surface of degree d = 4

In this case R is of type D_{5} and W_{S} ≅ 2^{4} : S_{5}. We use the following well-known classical result.

Lemma 6.5. Let S be a Del Pezzo surface of degree 4. Then S is isomorphic to a nonsingular surface of degree 4 in ℙ^{4} given by equations

F_{1} = ∑_{i=0}^{4} t_{i}^{2} = 0, F_{2} = ∑_{i=0}^{4} a_{i}t_{i}^{2} = 0,  (37)

where all a_{i}’s are distinct.

Proof. It is known that a Del Pezzo surface in its anti-canonical embedding is projectively normal. Using Riemann-Roch, one obtains that S is a complete intersection Q_{1} ∩ Q_{2} of two quadrics. Let P = λQ_{1} + μQ_{2} be the pencil spanned by these quadrics. The locus of singular quadrics in the pencil is a homogeneous equation of degree 5 in the coordinates λ, μ. Since S is nonsingular, it is not hard to see that the equation has no multiple roots (otherwise P contains a reducible quadric or there exists a quadric in the pencil with singular point at S, in both cases S is singular). Let p_{1}, …, p_{5} be the singular points of singular quadrics from the pencil. Suppose they are contained in a hyperplane H. Since no quadrics in the pencil contains H, the restriction P|H of the pencil of quadrics to H contains ≥ 5 singular members. This implies that all quadrics in P|H are singular. By Bertini’s theorem, all quadrics are singular at some point p ∈ H. This implies that all quadrics in P are tangent to H at p. One of the quadrics must be singular at p, and hence S is singular at p. This contradiction shows that p_{1}, …, p_{5} span ℙ^{4}. Choose coordinates in ℙ^{4} such that the singular points of singular quadrics from P are the points (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), and so on. Then each hyperplane V (t_{i}) = (t_{i} = 0) is a common tangent hyperplane of quadrics from P at the point p_{i}. This easily implies that the equations of quadrics are given by (37).

Let Q_{i} = V (a_{i}F_{1} − F_{2}), i = 0, …, 4, be one of the singular quadrics in the pencil P. It is a cone over a nonsingular quadric in ℙ^{3}, hence it contains two families of planes. The intersection of a plane with any other quadric in the pencil is a conic contained in S. Thus each Q_{i} defines a pair of pencils of conics |C_{i}| and |C′_{i}|, and it is easy to see that |C_{i} + C′_{i}| = | − K_{S}|.

Proposition 6.6. Let S be a Del Pezzo surface given by equations (37). The divisor classes c_{i} = [C_{i}] together with K_{S} form a basis of Pic(S) ⊗ ℚ. The Weyl group W_{S} acts on this basis by permuting the c_{i}’s and sending c_{i} to c′_{i} = [C′_{i}] = −K_{S} − c_{i}.

Proof. If we choose a geometric basis (e_{0}, e_{1}, …, e_{5}) in Pic(S), then the 5 pairs of pencils of conics are defined by the classes e_{0} − e_{i}, 2e_{0} − e_{1} − ⋯ − e_{5} + e_{i}. It is easy to check that the classes [C_{i}]’s and K_{S} form a basis in Pic(S)⊗ℚ. The group W_{S} contains a subgroup isomorphic to S_{5} generated by the reflections in vectors e_{1}−e_{2}, …, e_{4}−e_{5}., It acts by permuting e_{1}, …, e_{5}, hence permuting the pencils |C_{i}|. It is equal to the semi-direct product of S_{5} and the subgroup isomorphic to 2^{4} which is generated by the conjugates of the product s of two commuting reflections with respect to the vectors e_{0} − e_{1} − e_{2} − e_{3} and e_{4} − e_{5}. It is easy to see that s([C_{4}]) = [C′_{4}], s([C_{5}]) = [C′_{5}] and s([C_{i}]) = [C_{i}] for i ≠ 4, 5. This easily implies that W_{S} acts by permuting the classes [C_{i}] and switching even number of them to [C′_{i}].

Corollary 6.7. Let W (D_{5}) act in ℂ^{5} by permuting the coordinates and switching the signs of even number of coordinates. This linear representation of W (D_{5}) is isomorphic to the representation of W (D_{5}) on R_{S} ⊗ ℂ.

The group of projective automorphisms generated by the transformations which switch x_{i} to −x_{i} generates a subgroup H of Aut(S) isomorphic to 2^{4}.

We identify the group H with the linear space of subsets of even cardinality of the set J = {0, 1, 2, 3, 4}, or, equivalently, with the subspace of 𝔽_{2}^{J} of functions with support at a subset of even cardinality. We equip H with the symmetric bilinear form defined by the dot-product in 𝔽_{2}^{J}, or, equivalently, by (A, B) = #A∩B mod 2. We denote elements of H by i_{A}, where i_{A} is the characteristic function of A ⊂ J.

There are two kinds of involutions i_{A}. An involution of the first kind corresponds to a subset A of 4 elements. The set of fixed points of such an involution is a hyperplane section of S, an elliptic curve. The trace formula (36) gives that the the trace of i_{A} in Pic(S) is equal to −2. The corresponding conjugacy class in W_{5} is of type 4A_{1}. There are 5 involutions of the first kind. The quotient surface S/⟨i_{A}⟩ = Q is isomorphic to a nonsingular quadric. The map S → Q coincides with the map S → ℙ^{1} × ℙ^{1} that is given by the pencils |C_{i}| and |C′_{i}|.

Involutions of the second type correspond to subsets A of cardinality 2. The fixed-point set of such involution consists of 4 isolated points. This gives that the trace is equal to 2, and the conjugacy class is of type 2A_{1}. The quotient S/(i_{A}) is isomorphic to the double cover of ℙ^{2} branched along the union of two conics.

The subgroup of the Weyl group W (D_{5}) generated by involutions from the conjugacy class of type 2A_{1} is the normal subgroup 2^{4} in the decomposition W (D_{5}) ≅ 2^{4} : S_{5}. The product of two commuting involutions from this conjugacy class is an involution of type 4A_{1}. Thus the image of H in W_{S} is a normal subgroup isomorphic to 2^{4}.

Let G ≅ 2^{a} be a subgroup of 2^{4}. All cyclic groups G are not minimal. There are three kinds of subgroups H of order 4 in 2^{4}. A subgroup of the

first kind does not contain an involution of the first kind. An example is the group generated by i_{01}, i_{12}. The trace of its nonzero elements equal to 1. So this group is not minimal.

A subgroup of the second type contains only one involution of the first kind. An example is the group generated by i_{01}, i_{23}. The trace formula gives rank Pic(S)^{H} = 2. So it is also non-minimal.

A subgroup of the third kind contains two involutions of the first kind. For, example a group generated by i_{1234}, i_{0234}. It contains 2 elements with trace −3 and one element with trace 1. Adding up the traces we see the group is a minimal group. It is easy to see that S^{H} consists of 4 isolated points.

Now let us consider subgroups of order 8 of 2^{4}. They are parametrized by the same sets which parametrize involutions. A subgroup H_{A} corresponding to a subset A consists of involutions i_{B} such that #A∩B is even. The subsets A correspond to linear functions on 2^{4}. If #A = 2, say A = {0, 1}, we see that H_{A} contains the involutions i_{01}, i_{01ab}, i_{cd}, c, d ≠ 0, 1. Adding up the traces we obtain that these subgroups are minimal.

If #A = 4, say A = {1, 2, 3, 4}, the subgroup H_{A} consists of i_{1234} and i_{ab}, where a, b ≠ 0. Adding up the traces we obtain that H_{A} is not minimal.

Since 2^{4} contains a minimal subgroup, it is minimal itself.

Now suppose that the image G′ of G in S_{5} is non-trivial. The subgroup S_{5} of Aut(S) can be realized as the stabilizer of a set of 5 skew lines on S (corresponding to the basis vectors e_{1}, …, e_{5}). Thus any subgroup H of S_{5} realized as a group of automorphisms of S is isomorphic to a group of projective transformations of ℙ^{2} leaving invariant a set of 5 points. Since there is a unique conic through these points, the group is isomorphic to a finite group of PGL(2) leaving invariant a set of 5 distinct points. In section 4, we listed all possible subgroups of GL(2) and in section 5 we described their relative invariants. It follows that a subgroup leaves invariant a set of 5 distinct points if and only if it is one of the following groups C_{2}, C_{3}, C_{4}, C_{5}, S_{3}, D_{10}. The corresponding binary forms of degree 5 are projectively equivalent to the following binary forms:

• C_{2} : t_{0}(t_{0}^{2} − t_{1}^{2})(t_{0}^{2} + at_{1}^{2}), a ≠ −1, 0, 1;

• C_{4} : t_{0}(t_{0}^{2} − t_{1}^{2})(t_{0}^{2} + t_{1}^{2});

• C_{3}, S_{3} : t_{0}t_{1}(t_{0} − t_{1})(t_{0} − ϵ_{3}t_{1})(t_{0} − ϵ_{3}^{2}t_{1});

• C_{5}, D_{10} : (t_{0} − t_{1})(t_{0} − ϵ_{5}t_{1})(t_{0} − ϵ_{5}^{2}t_{1})(t_{0} − ϵ_{5}^{3}t_{1})(t_{0} − ϵ_{5}^{4}t_{1}).

The corresponding surfaces are projectively equivalent to the following surfaces

C_{2} : x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = x_{0}^{2} + ax_{1}^{2} − x_{2}^{2} − ax_{3}^{2} = 0, a ≠ −1, 0, 1(38)
C_{4} : x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = x_{0}^{2} + ix_{1}^{2} − x_{2}^{2} − ix_{3}^{2} = 0  (39)
S_{3} : x_{0}^{2} + ϵ_{3}x_{1}^{2} + ϵ_{3}^{2}x_{2}^{2} + x_{3}^{2} = x_{0}^{2} + ϵ_{3}^{2}x_{1}^{2} + ϵ_{3}x_{2}^{2} + x_{4}^{2} = 0  (40)
D_{10} : ∑_{i=0}^{4} ϵ_{5}^{i}x_{i}^{2} = ∑_{i=0}^{4} ϵ_{5}^{4−i}x_{i}^{2} = 0  (41)

Remark 6.8. Note that equations (40), (41), (41) are specializations of equation (39). It is obvious for equation (40) where we have to take a = i. Equation (39) specializes to equation (41) when we take a = ±\frac{1}{\sqrt{−3}} (use

that the Moebius transformation of order 3 x ↦ \frac{\sqrt{a}x+1}{x+\sqrt{a}} permutes cyclically ∞, \sqrt{a}, −\sqrt{−a} and fixes 1, −1). Equation (39) specializes to equation (41) if we take a = −2 ± \sqrt{5} (use that the Moebius transformation x ↦ \frac{x+2a−1}{x+1} permutes cyclically (∞, 1, a, −a, −1)). We thank J. Blanc for this observation.

Since the subgroup S_{5} leaves the class e_{0} invariant, it remains to consider subgroups G of 2^{4} : S_{5} which are not contained in 2^{4} and not conjugate to a subgroup of S_{5}. We use the following facts.

1) Suppose G contains a minimal subgroup of 2^{4}. Then G is minimal.

2) Let \overline{G} be the image of G in S_{5}. Then it is a subgroup of one of the groups listed above.

3) The group W (D_{5}) is isomorphic to the group of transformations of ℝ^{5} which consists of permutations of coordinates and changing even number of signs of the coordinates. Each element w ∈ W (D_{5}) defines a permutation of the coordinate lines which can be written as a composition of cycles (i_{1} … i_{k}).

If w changes signs of even number of the coordinates x_{i_{1}}, …, x_{i_{k}}, the cycle is called positive. Otherwise it is called a negative cycle. The conjugacy class of w is determined by the lengths of positive and negative cycles, except when all cycles of even length and positive in which case there are two conjugacy classes. The latter case does not occur in the case when n is odd. Assign to a positive cycle of length k the Carter graph A_{k−1}. Assign to a pair of negative cycles of lengths i ≥ j the Carter graph of type D_{i+1} if j = 1 and D_{i+j}(a_{j−1}) if j > 1. Each conjugacy class is defined by the sum of the graphs. We identify D_{2} with 2A_{1}, and D_{3} with A_{3}. In Table 2 below we give the conjugacy classes of elements in W (D_{5}), their characteristic polynomials and the traces in the root lattice of type D_{5}.

Order Notation Characteristic polynomial Trace  Representatives
2  A_{1}  t + 1  3  (ab)
2  2A_{1}  (t + 1)^{2}  1  (ab)(cd), (ab)(cd)i_{abcd}
2  2A_{1}^{∗}  (t + 1)^{2}  1  (ab)(cd), (ab)(cd)i_{ab}
2  3A_{1}  (t + 1)^{3}  -1  (ab)i_{cd}
2  4A_{1}  (t + 1)^{4}  -3  i_{abcd}
3  A_{2}  t^{2} + t + 1  2  (abc), (abc)i_{ab}
4  A_{3}  t^{3} + t^{2} + t + 1  1 (abcd), (abcd)i_{ab}, (abcd)i_{abcd}
4 A_{1} + A_{3} (t^{3} + t^{2} + t + 1)(t + 1) -1  (ab)(cd)i_{ae}
4 D_{4}(a_{1})  (t^{2} + 1)^{2}  1  (ab)(cd)i_{ac}
5  A_{4}  (t^{4} + t^{3} + t^{2} + t + 1)  0  (abcde), (abcde)i_{A}
6 A_{2} + A_{1}  (t^{2} + t + 1)(t + 1)  0  (ab)(cde)
6 A_{2} + 2A_{1}  (t^{2} + t + 1)(t + 1)^{2}  -2  (abc)i_{abde}, (abc)i_{de}
6  D_{4}  (t^{3} + 1)(t + 1)  0  (abc)i_{abce}
8  D_{5}  (t^{4} + 1)(t + 1) -1  (abcd)i_{abce}, (abcd)i_{de}
12 D_{5}(a_{1})  (t^{3} + 1)(t^{2} + 1)  0  (abc)(de)i_{ac}

Table 3. Conjugacy classes in W (D_{5})

In the following \overline{G} denotes the image of G in K = W (D_{5})/2^{4} ≅ S_{5}. Case 1. \overline{G} ≅ C_{2}.

It follows from the description of the image of Aut(S) in W (D_{5}) given in Corollary 6.7, that \overline{G} is generated by the permutation s = (02)(13). Let g ∉ G∩2^{4}. Then g = s or g = si_{A} for some A. It follows from Table 3 that g is either of type 2A_{1}, or of type A_{1}+A_{3}, or of type D_{4}(a_{1}). Let K = G∩2^{4} ≅ 2^{a}. If a = 0 or 1, the group is not minimal.

a = 2.

Suppose first that s acts identically on K. Then the group is commutative isomorphic to 2^{3} if it does not contain elements of order 4 and 2×4 otherwise. In the first case

K = F := ⟨i_{02}, i_{13}⟩.  (42)

is the subspace of fixed points of s in 2^{4}. Since (si_{A})^{2} = i_{A+s(A)}, we see that G \ K = {si_{A}, A ∈ K}. Consulting Table 3 we compute the traces of all elements from G to conclude that the total sum is equal to 8. Thus the group is not minimal.

In the second case G contains 4 elements of order 4 of the form si_{A}, where A ∉ K. Suppose A + s(A) = {0, 1, 2, 3}. Since K is a subspace of the second type, it contributes 4 to the total sum of the traces. Thus the sum of the traces of the elements of order 4 must be equal to −4. In other words they have to be elements with trace −1 of the form si_{A}, where #A = 2, 4 ∈ A. This gives the unique conjugacy class of a minimal group isomorphic to 2 × 4. It is represented by the group G = ⟨K, si_{04}⟩.

Assume now that G is non-abelian, obviously isomorphic to D_{8}. The subspace K contains one element from the set F. The nontrivial coset contains 2 elements of order 4 and two elements of order 2. Suppose K is of the second type with the sum of the traces of its elements equal to 4. Two elements of order 2 in G \ K have the trace equal to 1. Elements of order 4 have the trace equal to 1 or −1. So the group cannot be minimal. Thus K must be of the third type, the minimal one. This gives us the minimal group conjugate to the subgroup G = ⟨i_{1234}, i_{02}, si_{04}⟩ isomorphic to D_{8}.

a = 3.

There are three s invariant subspaces of 2^{4} of dimension 3. Their orthogonal complements are spanned by the one of the vectors in the set (42). As we saw earlier, if K^{⊥} = ⟨i_{A}⟩, where #A = 2, the subspace K is a minimal group. Otherwise the total sum of the traces of elements from K is equal to 8. In the first case we may assume that K = ⟨i_{14}, i_{34}, i_{02}⟩. All elements of order 2 in the nontrivial coset have the trace equal to 1. Thus we must have elements of order 4 in the coset with trace −1. Let si_{A} be such an element, where we may assume that A = {0, 4}. Thus G = ⟨K, si_{04}⟩. Its nontrivial coset has 4 elements of order 4 with trace 1 and 4 elements of order 4 with trace −1. The group is minimal. It is a non-split extension (2^{3})^{•}2. Its center is isomorphic to 2^{2}. The classification of groups of order 16 from Table 1 shows that this is group is isomorphic to L_{16}.

a = 4.

In this case G = 2^{4} : 2, where the extension is defined by the action of s in 2^{4}. The group has 2-dimensional center with the quotient isomorphic to 2^{3}.

Case 2. \overline{G} ≅ C_{3}.

We may assume that \overline{G} = ⟨s⟩, where s = (012). Applying Lemma 4.2, we obtain that G is a split extension K : 3, where K = G ∩ 2^{4} ≅ 2^{a}. Since there are no minimal elements of order 3, we must have a > 0. If a = 1, the group is 2 : 3 ≅ 6. There are no minimal elements of order 6, so we may assume that a > 1.

Assume a = 2. The group is abelian 2^{2} × 3 or non-abelian 2^{2} : 3 ≅ A_{4}. In the first case, K = ⟨i_{0123}, i_{0124}⟩ is the subspace of the third type, the minimal one. Thus the total number of elements in the nontrivial cosets is equal to 0. An element of order 3 has the trace equal to 1. An element of order 6 has trace equal to −2 of 0. So we must have an element of order 6 with trace −2. It must be equal to si_{34}. Its cube is i_{34} ∈ K. So we get one conjugacy class of a minimal group isomorphic to 2^{2} × 3. It is equal to ⟨K, si_{34}⟩.

If G ≅ A_{4}, the subspace K is not minimal. The group does not contain elements of order 6. So the traces of all elements not from K are positive. This shows that the group cannot be minimal.

Assume a = 3. The subspace K is minimal if and only if its orthogonal complement is generated by i_{34}. Again the group must contain si_{34} with trace −2 and hence equal to ⟨K, si_{34}⟩ = ⟨K, s⟩. The group G is minimal and is isomorphic to 2 × (2^{2} : 3) ≅ 2 × A_{4}.

Finally assume that K = ⟨i_{0123}⟩^{⊥} is not minimal. We have computed earlier the sum of the traces of its elements. It is equal to 8. Again it must contain an element of order 6 equals si_{0123}. Since K contains i_{0123}, the group contains s. Now we can add all the traces and conclude that the group is not minimal.

Of course we should not forget the minimal group 2^{4} : 3.

Case 3. \overline{G} ≅ S_{3}.

The group \overline{G} is generated by the permutations of coordinates (012) and (12)(34). It is immediately checked that H = G ∩ 2^{4} is not trivial for minimal G. The only subgroup of H invariant with respect to the conjugation action of \overline{G} on H is H itself. This gives a minimal group isomorphic to 2^{4} : S_{3}. The extension is defined by the restriction to S_{3} of the natural action of S_{4} = W (A_{4}) on its root lattice modulo 2.

Case 4. \overline{G} ≅ C_{4}.

The group 2^{4} : 4 contains 2^{4} : 2, so all minimal groups of the latter group are minimal subgroups of 2^{4} : 4. Without loss of generality, we may assume that the group \overline{G} is generated by the permutations of coordinates s = (0123). The only proper subgroup of 2^{4} invariant with respect to the conjugation action of \overline{G} on K = G ∩ 2^{4} is either ⟨i_{0123}⟩ or its orthogonal complement. In the first case G ≅ 2•4 ≅ 2 × 4 or 8. In the first case the group is not minimal. In the second case G = ⟨(0123)i_{0123}⟩ is minimal.

Assume K = ⟨i_{0123}⟩^{⊥}. If s ∉ G, then si_{A} ∈ G, where A ∉ K. The Table 3 shows that all such elements are minimal of order 8. This gives a minimal group G ≅ 2^{3} : 4 = 2^{2} : 8.

Next we have to consider the case when s ∈ G so that G = ⟨K, s⟩. The total number of traces of elements from K is equal to 8. Consulting the Table 3 we obtain that the elements in the cosets sK, s^{2}K, s^{3}K have the trace equal to 1. So the group is not minimal.

Our last minimal group in this case is 2^{4} : 4.

Case 5. \overline{G} = C_{5} or D_{10}.

Again, we check using the table of conjugacy classes that no group isomorphic to D_{10} is minimal. Also no proper subgroup of H is invariant with respect to conjugation by a permutation of order 5, or by a subgroup of S_{5} generated by (012) and (12)(34). Thus we get two minimal groups isomorphic to 2^{4} : 5 or 2^{4} : D_{10}.

The following theorem summarizes what we have found.

Theorem 6.9. Let (S, G) be a minimal Del Pezzo surface of degree d = 4. Then G is isomorphic to one of the following groups

1. Aut(S) ≅ 2^{4}.

2^{4}, 2^{3}, 2^{2}.

2. Aut(S) ≅ 2^{4} : 2

2 × 4, D_{8}, L_{16}, 2^{4} : 2,

and from the previous case.

3. Aut(S) ≅ 2^{4} : 4

8, 2^{2} : 8, 2^{4} : 4,

and from the previous two cases.

4. Aut(S) ≅ 2^{4} : S_{3}.

2^{2} × 3, 2 × A_{4}, 2^{4} : 3, 2^{4} : S_{3},

and from Cases 1) and 2).

5. Aut(S) ≅ 2^{4} : D_{10}

2^{4} : D_{10}, 2^{4} : 5,

and from Cases 1) and 2).

6.5 Cubic surfaces

The following theorem gives the classification of cyclic subgroups of Aut(S) and identifies the conjugacy classes of their generators.

Theorem 6.10. Let S be a nonsingular cubic surface admitting a non-trivial automorphism g of order n. Then S is equivariantly isomorphic to one of the following surfaces V (F) with

g = [x_{0}, ϵ_{n}^{a}x_{1}, ϵ_{n}^{b}x_{2}, ϵ_{n}^{c}x_{3}].  (43)

• 4A_{1} (n = 2), (a, b, c) = (0, 0, 1),

F = T_{3}^{2}L_{1}(T_{0}, T_{1}, T_{2}) + T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + αT_{0}T_{1}T_{2}.

• 2A_{1} (n = 2), (a, b, c) = (0, 1, 1),

F = T_{0}T_{2}(T_{2} + αT_{3}) + T_{1}T_{3}(T_{2} + βT_{3}) + T_{0}^{3} + T_{1}^{3}.

• 3A_{2} (n = 3), (a, b, c) = (0, 0, 1),

F = T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + αT_{0}T_{1}T_{2}.

Finite subgroups of the plane Cremona group  491

• A_{2} (n = 3), (a, b, c) = (0, 1, 1), F = T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3}.

• 2A_{2} (n = 3), (a, b, c) = (0, 1, 2), F = T_{0}^{3} + T_{1}^{3} + T_{2}T_{3}(T_{0} + aT_{1}) + T_{2}^{3} + T_{3}^{3}.

• D_{4}(a_{1}) (n = 4), (a, b, c) = (0, 2, 1), F = T_{3}^{2}T_{2} + L_{3}(T_{0}, T_{1}) + T_{2}^{2}(T_{0} + αT_{1}).

• A_{3} + A_{1} (n = 4), (a, b, c) = (2, 1, 3), F = T_{0}^{3} + T_{0}T_{1}^{2} + T_{1}T_{3}^{2} + T_{1}T_{2}^{2}.

• A_{4} (n = 5), (a, b, c) = (4, 1, 2), F = T_{0}^{2}T_{1} + T_{1}^{2}T_{2} + T_{2}^{2}T_{3} + T_{3}^{2}T_{0}.

• E_{6}(a_{2}) (n = 6), (a, b, c) = (0, 3, 2), F = T_{0}^{3} + T_{1}^{3} + T_{3}^{3} + T_{2}^{2}(αT_{0} + T_{1}).

• D_{4} (n = 6), (a, b, c) = (0, 2, 5), F = L_{3}(T_{0}, T_{1}) + T_{3}^{2}T_{2} + T_{2}^{3}.

• A_{5} + A_{1} (n = 6), (a, b, c) = (4, 2, 1), F = T_{3}^{2}T_{1} + T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + λT_{0}T_{1}T_{2}.

• 2A_{1} + A_{2} (n = 6), (a, b, c) = (4, 1, 3), F = T_{0}^{3} + βT_{0}T_{3}^{2} + T_{2}^{2}T_{1} + T_{1}^{3}.

• D_{5} (n = 8), (a, b, c) = (4, 3, 2), F = T_{3}^{2}T_{1} + T_{2}^{2}T_{3} + T_{0}T_{1}^{2} + T_{0}^{3}.

• E_{6}(a_{1}) (n = 9), (a, b, c) = (4, 1, 7), F = T_{3}^{2}T_{1} + T_{1}^{2}T_{2} + T_{2}^{2}T_{3} + T_{0}^{3}.

• E_{6} (n = 12), (a, b, c) = (4, 1, 10), F = T_{3}^{2}T_{1} + T_{2}^{2}T_{3} + T_{0}^{3} + T_{1}^{3}.

We only sketch a proof, referring for the details to [26]. Let g be a nontrivial projective automorphism of S = V (F) of order n. All possible values of n can be obtained from the classification of conjugacy classes of W (E_{6}). Choose coordinates to assume that g acts as in (43). Then F is a sum of monomials which belong to the same eigensubspace of g in its action in the space of cubic polynomials. We list all possible eigensubspaces. Since V (F) is nonsingular, the square or the cube of each variable divides some monomial entering in F. This allows one to list all possible nonsingular V (F) admitting an automorphism g. Some obvious linear change of variables allows one to find normal forms. Finally, we determine the conjugacy class by using the trace formula (36) applied to the locus of fixed points of g and its powers.

The conjugacy class labeled by the Carter graph with 6 vertices defines a minimal cyclic group.

Corollary 6.11. The following conjugacy classes define minimal cyclic groups of automorphisms of a cubic surface S.

• 3A_{2} of order 3,

• E_{6}(a_{2}) of order 6,

• A_{5} + A_{1} of order 6,

• E_{6}(a_{1}) of order 9,

• E_{6} of order 12.

Next we find all possible automorphism groups of nonsingular cubic surfaces. Using a normal form of a cubic admitting a cyclic group of automorphisms from given conjugacy class, we determine all other possible symmetries of the equation. We refer for the details to [26]. The list of possible automorphism groups of cubic surfaces is given in Table 4.

Remark 6.12. Note that there are various ways to write the equation of cubic surfaces from the table. For example, using the identity

(x + y + z)^{3} + ϵ_{3}(x + ϵ_{3}y + ϵ_{3}^{2}z)^{3} + ϵ_{3}^{2}(x + ϵ_{3}^{2}y + ϵ_{3}z)^{3} = 9(x^{2}z + y^{2}x + z^{2}y)

we see that the Fermat cubic can be given by the equation

T_{0}^{3} + T_{1}^{2}T_{3} + T_{3}^{2}T_{2} + T_{2}^{2}T_{1} = 0.

Using Theorem 6.10 this exhibits a symmetry of order 9 of the surface, whose existence is not obvious in the original equation.

Using the Hesse form of an equation of a nonsingular plane cubic curve we see that a surface with equation

T_{0}^{3} + F_{3}(T_{1}, T_{2}, T_{3}) = 0

is projectively equivalent to a surface with the equation

T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + 6aT_{0}T_{1}T_{2} = 0.

Type Order Structure  F (T_{0}, T_{1}, T_{2}, T_{3})  Parameters
I  648 3^{3} : S_{4}  T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3}
II  120  S_{5}  T_{0}^{2}T_{1} + T_{0}T_{2}^{2} + T_{2}T_{3}^{2} + T_{3}T_{1}^{2}
III  108 H_{3}(3) : 4  T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + 6aT_{1}T_{2}T_{3} 20a^{3} + 8a^{6} = 1
IV  54 H_{3}(3) : 2  T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + 6aT_{1}T_{2}T_{3}  a − a^{4} ≠ 0,

8a^{3} ≠ −1, 20a^{3} + 8a^{6} ≠ 1

V 24 S_{4} T_{0}^{3} + T_{0}(T_{1}^{2} + T_{2}^{2} + T_{3}^{2}) 9a^{3} ≠ 8a
+aT_{1}T_{2}T_{3}  8a^{3} ≠ −1,
VI  12 S_{3} × 2 T_{2}^{3} + T_{3}^{3} + aT_{2}T_{3}(T_{0} + T_{1}) + T_{0}^{3} + T_{1}^{3}  a ≠ 0
VII  8  8  T_{3}^{2}T_{2} + T_{2}^{2}T_{1} + T_{0}^{3} + T_{0}T_{1}^{2}
VIII  6  S_{3} T_{2}^{3} + T_{3}^{3} + aT_{2}T_{3}(T_{0} + bT_{1}) + T_{0}^{3} + T_{1}^{3}  a^{3} ≠ −1
IX  4  4  T_{3}^{2}T_{2} + T_{2}^{2}T_{1} + T_{0}^{3} + T_{0}T_{1}^{2} + aT_{1}^{3}  a ≠ 0
X  4  2^{2}  T_{0}^{2}(T_{1} + T_{2} + aT_{3}) + T_{1}^{3} + T_{2}^{3}
+T_{3}^{3} + 6bT_{1}T_{2}T_{3}  8b^{3} ≠ −1
XI 2 2 T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + 6aT_{1}T_{2}T_{3} b^{3}, c^{3} ≠ 1
+T_{0}^{2}(T_{1} + bT_{2} + cT_{3})  b^{3} ≠ c^{3}

8a^{3} ≠ −1,

Table 4. Groups of automorphisms of cubic surfaces

The special values of the parameters a = 0, 1, ϵ_{3}, ϵ_{3}^{2} give the Fermat cubic. The values a satisfying 20a^{3} +8a^{6} = 1 give a plane cubic with an automorphism of order 4 (a harmonic cubic). Since a harmonic cubic is isomorphic to the cubic with equation T_{1}^{3} + T_{1}T_{2}^{2} + T_{3}^{3} = 0, using Theorem 6.10 we see symmetries of order 6 from the conjugacy class E_{6}(a_{2}) for surfaces of type III, IV and of order 12 for the surface

T_{3}^{2}T_{1} + T_{2}^{2}T_{3} + T_{0}^{3} + T_{1}^{3} = 0

of type III.

It remains to classify minimal groups G. Note that if G is realized as a group of projective (or weighted projective) automorphisms of a family of surfaces (S_{t}), then G is a subgroup of the group of projective automorphisms of any surface S_{t_{0}} corresponding to a special value t_{0} of the parameters. We indicate this by writing S′ → S. The types of S′ when it happens are

IV → III, IV → I, V I, V III, IX → I, XI → X.

So it suffices to consider the surfaces of types I, II, III, V, VII, X.

We will be using the following lemma, kindly communicated to us by R. Griess. For completeness sake, we provide its proof.

Lemma 6.13. Let S_{n+1} act naturally on its root lattice R_{n} = {(a_{1}, …, a_{n+1}) ∈ ℤ^{n+1} : a_{1} + ⋯ + a_{n+1} = 0}. Let R_{n}(p) ≅ 𝔽_{p}^{n} be the reduction of R_{n} modulo a prime number p > 2 not dividing n + 1. Then the set of conjugacy classes of subgroups dividing a splitting of 𝔽_{p}^{n} : S_{n+1} is bijective to the set 𝔽_{p} if p | n + 1 and consists of one element otherwise.

Proof. It is easy to see that, fixing a splitting, there is a natural bijection between conjugacy classes of splitting subgroups in A : B and the cohomology set H^{1}(B, A), where B acts on A via the homomorphism B → Aut(A) defining the semi-direct product. So, it suffices to prove that H^{1}(S_{n+1}, R_{n}(p)) ≅ 𝔽_{p} if p | n+1 and zero otherwise. Consider the permutation representation of S_{n+1} on M = 𝔽_{p}^{n+1}. We have an exact sequence of S_{n+1}-modules

0 → R_{n}(p) → M → 𝔽_{p} → 0

defined by the map (a_{1}, …, a_{n+1}) → a_{1} + ⋯ + a_{n+1}. The module M is the induced module of the trivial representation of the subgroup S_{n} of S_{n+1}. By Eckmann-Shapiro’s Lemma,

H^{1}(S_{n+1}, M) = H^{1}(S_{n}, 𝔽_{p}) = Hom(S_{n}, 𝔽_{p}).

Suppose p ∤ n + 1, then the exact sequence splits, and we get

0 = H^{1}(S_{n+1}, M) = H^{1}(S_{n+1}, R_{n}(p)) ⊕ H^{1}(S_{n+1}, 𝔽_{p}).

Since H^{1}(S_{n+1}, 𝔽_{p}) = Hom(S_{n+1}, 𝔽_{p}) = 0, we get the result. If p | n + 1, then H^{0}(S_{n+1}, M) = 0, H^{0}(S_{n+1}, 𝔽_{p}) = 𝔽_{p} and the exact sequence of cohomology gives the desired result.

Type I.

Let us first classify 𝔽_{3}-subspaces of the group K = 3^{3}. We view 3^{3} as the S_{4}-module R_{3}(3) ≅ 𝔽_{3}^{3} from the previous lemma. We denote the image of a vector (a, b, c, d) in K by [a, b, c, d]. In our old notations

[a, b, c, d] = [ϵ^{a}x_{0}, ϵ^{b}x_{1}, ϵ^{c}x_{2}, ϵ^{d}x_{3}].

There are 13 (= #ℙ^{2}(𝔽_{3})) one-dimensional subspaces in 3^{3}. The group S_{4} acts on this set with 3 orbits. They are represented by vectors [1, 2, 0, 0], [1, 1, 1, 0], [1, 1, 2, 2] with respective stabilizer subgroups 2^{2}, S_{3} and D_{8}. We call them lines of type I,II,III, respectively. As subgroups they are cyclic groups of order 3 of the following types.

2A_{2} Type I,
3A_{2} Type II,
A_{2} Type III.

The conjugacy class of a 2-dimensional subspace K is determined by its orthogonal complement in 3^{3} with respect to the dot-product pairing on 𝔽_{3}^{4}. Thus we have 3 types of 2-dimensional subspaces of types determined by the type of its orthogonal complement.

An easy computation gives the following table.

Type 3A_{2} 2A_{2} A_{2} Trace
I 4 2 2 0
II 2 6 0 0
III 0 4 4 2

(44)

Here we list the types of the nontrivial elements in the subspace and the last column is the sum \frac{1}{9} ∑_{g∈L} Tr(g|R_{S}). This gives us two conjugacy classes of minimal subgroups isomorphic to 3^{2}.

Let G be a subgroup of Aut(S), \overline{G} be its image in S_{4} and K = G ∩ 3^{3}. Let k = dim_{𝔽_{3}} K.

Case 1 : k = 0.

In this case G defines a splitting of the projection 3^{3} : S_{4} → S_{4}. Assume G ≅ S_{4}. It follows from Lemma 6.13 that there are 3 conjugacy classes of subgroups isomorphic to S_{4} which define a splitting. Let us show each of them is minimal.

We start with the standard S_{4} generated by permutations of the coordinates. It contains 6 elements of type 4A_{1}, 8 elements of type 2A_{2}, 3 elements of type 2A_{1} and 6 elements of type A_{3} + A_{1}. Adding up the traces we obtain that the group is minimal.

Suppose G is another subgroup isomorphic but not conjugate to the previous S_{4}. It corresponds to a 1-cocylce ϕ : S_{4} → 𝔽_{3}^{3} defined by a vector v = (a_{1}, a_{2}, a_{3}, a_{4}) with a_{1} + a_{2} + a_{3} + a_{4}

ϕ_{v}(σ) = (a_{1}, a_{2}, a_{3}, a_{4}) − (a_{σ(1)}, a_{σ(2)}, a_{σ(3)}, a_{σ(4)}) = (a, b, c, d).

The cohomology class of this cocycle depends only on the sum of the coordinates of the vector v. Without loss of generality we may choose v = (1, 0, 0, 0) and drop the subscript in ϕ. We identify the vector ϕ(σ) with the automorphism [ϵ_{3}^{a}x_{0}, ϵ_{3}^{b}x_{1}, ϵ_{3}^{c}x_{2}, ϵ_{3}^{d}x_{3}]. Thus a new S_{4} consists of transformation σ ∘ ϕ(σ). We check that the type of an element σϕ(σ) is equal to the type of σ for each σ ∈ S_{4}. This shows that the representation of a new S_{4} in R_{S} has the same character as that of the old S_{4}. This shows that all three S_{4}’s are minimal.

Let G be mapped isomorphically to a subgroup G′ of S_{4}. If G′ is a 2-group, then it is contained in a 2-Sylow subgroup of one of the S_{4}’s. By the above its representation in R_{S} is the same as the restriction of the representation of S_{4}. A 2-Sylow subgroup of S_{4} contains 3 elements of type 2A_{1}, two element of type 4A_{1} and 2 elements of type A_{3} + A_{1}. Adding up the traces we get 8. Thus the group is not minimal.

If G′ is not a 2-group, then G′ is either S_{3} or 3. A lift of a permutation (i23) is given by a matrix

(\begin{matrix}0 & 0 & ϵ_{3}^{a} \\ ϵ_{3}^{b} & 0 & 0 \\ 0 & ϵ_{3}^{c} & 0 \end{matrix}) ,

where a + b + c ≡ 0 mod 3. It is immediately checked that all such matrices define an element of type 2A_{2}. Adding up the traces we see that a lift of S_{3} is minimal.

Case 2 : k = 1.

Let G′ be the image of G in S_{4} and K = G ∩ 3^{3}. Clearly, the subspace K must be invariant with respect to the restriction of the homomorphism S_{4} → Aut(3^{3}) to G′.

Assume K is of type I, say generated by n_{K} = [1, 2, 0, 0]. Then its stabilizer in S_{4} is generated by (12), (34), (123) isomorphic to 2^{2}. The conjugation by (12) sends n_{K} to −n_{K}, and the conjugation by (34) fixes n_{K}. Thus the group K : 2^{2} is isomorphic S_{3} × 2. It is easy to check that the product g = (34)n_{K} is of order 6, of minimal type A_{5} + A_{1}. Thus the groups K : 2^{2} and its cyclic subgroup of order 6 are minimal. Also its subgroup of order 6 is minimal. Its subgroup ⟨n_{K}, (12)⟩ ≅ S_{3} contains 3 elements of type 4A_{1} and 2 elements of type 2A_{2}. Adding up the traces we see that S_{3} is also minimal. It is obviously not conjugate from the S_{3} from the previous case.

Assume K is of type II, say generated by n_{K} = [1, 1, 1, 0]. Since n_{K} is minimal, any subgroup in this case is minimal. The stabilizer of K in S_{4} is generated by (123), (12) and is isomorphic to S_{3}. Our group G is a subgroup of 3•S_{3} with K contained in the center. There are three non-abelian groups of order 18: D_{18}, 3 × S_{3}, 3^{2} : 2. The extension in the last group is defined by the automorphism of 3^{2} equal to the minus identity. In our case the image of G in S_{3} acts identically on K. Since the center of D_{18} or 3^{2} : 2 is trivial, this implies that either G is a cyclic subgroup of D_{18} of order 9 or 3, or a subgroup of S_{3} × 3 in which case G ≅ 3, 3^{2}, 6, 3 × S_{3}. Note that the group 3^{2} is not conjugate to a subgroup of 3^{3}. To realize a cyclic subgroup of order 9 is enough to take g = n_{K}(234). Note that the Sylow subgroup of 3•S_{3} is of order 9, so all 3-subgroups of order 9 are conjugate.

Assume K is of type III, say generated by n_{K} = [1, 1, 2, 2]. The stabilizer group is generated by (12), (34), (13)(24) and is isomorphic to D_{8}. Our group G is a subgroup of 3 : D_{8}. The split extension is defined by the homomorphism D_{8} → 2 with kernel ⟨(12), (34)⟩ ≅ 2^{2}. The subgroup isomorphic to D_{8} is contained in a non-minimal S_{4}, hence is not minimal. Let H = ⟨n_{K}, (12), (34)⟩ ≅ 6×2 so that 3 : D_{8} ≅ H : 2. The subgroup H contains 4 elements of type D_{4}, 4 elements of type A_{2}, 2 elements of type A_{1} and one element of type 2A_{1}. Adding up the traces of elements from H we get the sum equal to 24. The nontrivial coset contains 8 elements of type A_{3} + A_{1}, 4 elements of type 4A_{1} and one element of type 2A_{1}. Adding up the traces we get 0. Thus the group is not minimal. So this case does not reveal any minimal groups.

Case 3 : k = 2.

The image of G in S_{4} is contained in the stabilizer of the orthogonal vector n_{K}. Thus G is a subgroup of one of the following groups

G = \begin{cases}K : 2^{2} & if K is of type I, \\ K•S_{3} & if K is of type II, \\ K : D_{8} & if K is of type III. \end{cases}

Since K of type I or II contains a minimal element of order 3, the subgroups of K : 2^{2} and K•S_{3} are minimal. Recall that they all contain K.

Assume K is of type I. Recall that the S_{4}-module 3^{3} is isomorphic to the root lattice R_{3}(3) of A_{3} modulo 3. A permutation σ of order 2 represented by the transposition (12) decomposes the module into the sum of one-dimensional subspaces with eigenvalues −1, 1, 1. So, if σ fixes n_{K} it acts on K with eigenvalues −1, 1. Otherwise it acts identically on K. The product (12)(34) acts with eigenvalues (−1, −1, 1), so if it fixes n_{K} then it acts as the minus identity on K. In our case n_{K} = [1, 2, 0, 0] and (12), (12)(34) ∈ 2^{2} fix n_{K}. Accordingly,

(12) acts as (−1, 1) giving a subgroup K : 2 ≅ 3 × S_{3}, (12)(34) gives the subgroup K : 2 ≅ 3^{2} : 2 ≅ 3̸^{2} : 2. Finally, (34) acts identically on K giving the subgroup 3^{2} × 2. So we obtain 3 subgroups of index 2 of K : 2^{2} isomorphic to 3^{2} : 2, 3^{2}×2, S_{3}×3. The remaining subgroups are K ≅ 3^{2} and K : 2^{2} ≅ 3^{2} : 2^{2}.

Assume K is of type II. Again we have to find all subgroups H of G = K•S_{3} containing K. Elements of order 2 are transpositions in S_{3}. They fix n_{K}. Arguing as above we see that K : 2 ≅ 3 × S_{3}. An element of order 3 in S_{3} fixes n_{K}. Hence it acts in the orthogonal complement as an element of type A_{2} in the root lattice of type A_{2} modulo 3. This defines a unique non-abelian group of order 27 isomorphic to the Heisenberg group H_{3}(3). The third group is K•S_{3} ≅ H_{3}(2) : 2 ≅ 3(3^{2} : 2).

Assume K is of type III. This time K is not minimal. Each subgroup of order 2 of type 4A_{1} of D_{8} defines a subgroup K : 2 ≅ 3 × S_{3} of K : D_{8}. It contains 3 elements of order 2, of type 4A_{1}, 6 elements of order 6, of type D_{4}, and the elements from K. Adding up the traces we get the sum equal to 18. So the subgroup is not minimal. An element of order 2 of type 2A_{1} defines a subgroup 3^{2} : 2 not isomorphic to 3 × S_{3}. It contains 9 elements of type 2A_{1}, and the elements from K. Adding up the traces we get the sum equal to 36. So the group is not minimal.

Assume G ≅ K : 4. It contains the previous group 3^{2} : 2 as a subgroup of index 2. It has 9 2-Sylow subgroups of order 4. Thus the nontrivial coset consists of 18 elements of order 4. Each element of order 4 has the trace equal tom 0. This shows that the group is not minimal.

Finally it remains to investigate the group K : D_{8}. It contains the previous group K : 4 as a subgroup of index 2. The sum of the traces of its elements is equal to 36. The nontrivial coset consists of the union of 4 subsets, each is equal to the set of nontrivial elements in the group of type K : 2 ≅ 3 × S_{3}. The sum of traces of elements in each subset is equal to 12. So the total sum is 72 and the group is not minimal.

Case 3 : k = 3.

This gives the groups 3^{3} : H, where H is a subgroup of S_{4} which acts on 3^{3} via the restriction of the homomorphism S_{4} → Aut(3^{3}) describing the action of W (A_{3}) on its root lattice modulo 3. We have the groups

3^{3} : S_{4}, 3^{3} : D_{8}, 3^{3} : S_{3}, 3^{3} : 2^{2}(2), 3^{3} : 3, 3^{3} : 4, 3^{3} : 2(2).

Type II.

The surface is isomorphic to the Segre cubic surface in ℙ^{4} given by the equations

∑_{i=0}^{4} x_{i}^{3} = ∑_{i=0}^{4} x_{i} = 0.

The group S_{5} acts by permuting the coordinates. The orbit of the line x_{0} = x_{1} + x_{2} = x_{3} + x_{4} = 0 consists of 15 lines. It is easy to see that the remaining 12 lines form a double-six. The lines in the double-six are described as follows.

Let ω be a primitive 5th root of unity. Let σ = (a_{1}, …, a_{5}) be a permutation of {0, 1, 2, 3, 4}. Each line ℓ_{σ} is the span by a pair of points (ω^{a_{1}}, …, ω^{a_{5}}) and (ω^{−a_{1}}, …, ω^{−a_{5}}). This gives 12 different lines. One immediately checks that ℓ_{σ} ∩ ℓ_{σ′} ≠ ∅ if and only if σ′ = σ ∘ τ for some odd permutation τ. Thus the orbit of the alternating subgroup A_{5} of any line defines a set of 6 skew lines (a sixer) and therefore A_{5} is not minimal. Let ℓ_{1}, …, ℓ_{6} be a sixer. It is known that the divisor classes ℓ_{i}, K_{S} span Pic(S) ⊗ ℚ. This immediately implies that Pic(S)^{A_{5}} is spanned (over ℚ) by K_{S} and the sum ∑ ℓ_{i}. Since S_{5} does not leave this sum invariant, we see that S_{5} is a minimal group.

A maximal subgroup of S_{5} not contained in A_{5} is isomorphic to S_{4}, or 5 : 4, or 2 × S_{3}. The subgroups isomorphic to S_{4} are conjugate so we may assume that it consists of permutations of 1, 2, 3, 4. The group has 6 elements of type 4A_{1} conjugate to (12), 3 elements of type 2A_{1} conjugate to (12)(34), 8 elements of type 2A_{2} conjugate to (123) and 6 elements of type A_{3} + A_{1} conjugate to (1234). The total sum of the traces is equal to 0. So the group is minimal. This gives another, non-geometric proof of the minimality of S_{5}.

Consider a 2-Sylow subgroup G of S_{4} isomorphic to D_{8}. It consists of 5 elements of order 2, two of type 4A_{1} (from the conjugacy class of (12)) and 3 of type 2A_{1} (from the conjugacy class of (12)(34)). Its cyclic subgroup of order 4 is generated by an element of type A_{3} + A_{1}. Adding up the traces we see that the subgroup is not minimal. Thus S_{4} has no minimal proper subgroups.

A subgroup isomorphic to 5 : 4 is conjugate to a subgroup generated by two cycles (01234) and (0123). Computing the traces, we find that the group is not minimal. The subgroup isomorphic to 2×S_{3} is conjugate to a subgroup generated by (012), (01), (34). Its element of order 6 belongs to the conjugacy class D_{4}. So this group is different from the isomorphic group in the previous case. Computing the traces we find that it is not minimal.

Type III.

The surface is a specialization of a surface of type IV. Recall that each nonsingular plane cubic curve is isomorphic to a member of the Hesse pencil

T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + 6aT_{1}T_{2}T_{3} = 0.

The group of projective automorphisms leaving the pencil invariant is the Hesse group G_{216} of order 216. It is isomorphic to 3^{2} : \overline{T}. The stabilizer of a general member of the pencil is isomorphic to a non-abelian extension 3^{2} : 2. It is generated by

g_{1} = [x_{1}, ϵ_{3}x_{2}, ϵ_{3}^{2}x_{3}], g_{2} = [x_{2}, x_{3}, x_{1}], g_{3} = [x_{2}, x_{1}, x_{3}].

The pencil contains 6 members isomorphic to a harmonic cubic. They correspond to the values of the parameters satisfying the equation 8a^{6} +20a^{3} −1 = 0. The stabilizer of a harmonic member is the group 3^{2} : 4. The additional generator is given by the matrix

g_{4} = \frac{1}{\sqrt{3}} (\begin{matrix}1 & 1 & 1 \\ 1 & ϵ_{3} & ϵ_{3}^{2} \\ 1 & ϵ_{3}^{2} & ϵ_{3} \end{matrix})

The pencil also contains 4 anharmonic cubics isomorphic to the Fermat cubic. They correspond to the parameters a satisfying the equation a^{4} − a = 0. The stabilizer of an anharmonic member is isomorphic to 3^{2} : 6. The additional generator is given by g_{5} = [x_{1}, ϵ_{3}x_{2}, ϵ_{3}x_{3}].

All curves from the Hesse pencil have 9 common inflection points. If we fix one of them, say (1, −1, 0), all nonsingular members acquire a group law. The group of automorphisms generated by g_{1}, g_{2} correspond to translations by 3-torsion points. The automorphism g_{3} is the negation automorphism. The automorphism g_{4} is the complex multiplication by \sqrt{−1}. The automorphism g_{5} is the complex multiplication by e^{2πi/3}.

The Hesse group admits a central extension 3G_{216} ≅ H_{3}(3) : \overline{T} realized as the complex reflection group in ℂ^{3}. It acts linearly on the variables T_{1}, T_{2}, T_{3} leaving the polynomial T_{1}^{3} + T_{2}^{3} + T_{3}^{3} + 6aT_{1}T_{2}T_{3} unchanged. We denote by g̃_{i} the automorphism of the cubic surface obtained from the automorphism g_{i} of the Hesse pencil by acting identically on the variable T_{0}. The center of the group 3G_{216} is generated by c = [g̃_{1}, g̃_{2}] = [1, ϵ_{3}, ϵ_{3}, ϵ_{3}]. This is an element of order 3, of minimal type 3A_{2}.

Now we have a complete description of the automorphism group of a surface of type IV. Any minimal subgroup of H_{3}(3) : 2 can be found among minimal subgroups of surfaces of type I. However, we have 2 non-conjugate subgroups of type S_{3} equal to ⟨g̃_{1}, g̃_{3}⟩ and ⟨g̃_{2}, g̃_{3}⟩, and two non-conjugate subgroups in S_{3} × 3 obtained from the previous groups by adding the central element c.

Surfaces of type III acquire additional minimal subgroups of the form A : 4, where A is a subgroup of H_{3}(3). The element g̃_{4} acts by conjugation on the subgroup H_{3}(3) via (g̃_{1}, g̃_{2}) ↦ (g̃_{2}^{2}, g̃_{1}). Using g̃_{4}, we can conjugate the subgroups isomorphic to S_{3}, 3 × S_{3}, 3^{2}. Also we get two new minimal groups H_{3}(3) : 4 and 12.

Type V.

The group S_{4} ≅ 2^{2} : S_{3} acts by permuting the coordinates T_{1}, T_{2}, T_{3} and multiplying them by −1 leaving the monomial T_{1}T_{2}T_{3} unchanged. To make the action explicit, we identify 2^{2} with the subspace of 𝔽_{2}^{3} of vectors whose coordinates add up to 0. The semi-direct product corresponds to the natural action of S_{3} by permuting the coordinates. Thus g = ((a, b, c), σ) ∈ 2^{2} : S_{3} acts as the transformation [x_{0}, (−1)^{a}x_{σ(1)}, (−1)^{b}x_{σ(2)}, (−1)^{c}x_{σ(3)}]. It is easy to compute the types of elements of S_{4} in their action on S. The group contains 3 elements of type 2A_{1}, 6 elements of type 4A_{1}, 8 elements of type 2A_{2} and 6 elements of type A_{3} + A_{1}. Adding up the traces we see that the group is minimal. The subgroup S_{3} is minimal. No other subgroup is minimal.

Type VII.

The automorphism group of the surface of type VII is a non-minimal cyclic group of order 8.

Type X.

The automorphism group of the surface of type X consists of the identity, two involutions of type 4A_{1} and one involution of type 2A_{1}. Adding up the traces, we get that the group is not minimal.

Let us summarize our result in the following.

Theorem 6.14. Let G be a minimal subgroup of automorphisms of a nonsingular cubic surface. Then G is isomorphic to one of the following groups.

1. G is a subgroup of automorphisms of a surface of type I.

S_{4}(3), S_{3} (2), S_{3} × 2, S_{3} × 3 (2), 3^{2} : 2 (2), 3^{2} : 2^{2},

H_{3}(3) : 2, H_{3}(3), 3^{3} : 2 (2), 3^{3} : 2^{2} (2), 3^{3} : 3, 3^{3} : S_{3}, 3^{3} : D_{8}, 3^{3} : S_{4}, 3^{3} : 4,

3^{3}, 3^{2} (3), 3^{2} × 2, 9, 6 (2), 3.

2. G is a subgroup of automorphisms of a surface of type II.

S_{5}, S_{4}.

3. G is a subgroup of automorphisms of a surface of type III.

H_{3}(3) : 4, H_{3}(3) : 2, H_{3}(3), S_{3} × 3, S_{3}, 3^{2}, 12, 6, 3.

4. G is a subgroup of automorphisms of a surface of type IV.

H_{3}(3) : 2, H_{3}(3), S_{3} (2), 3 × S_{3} (2), 3^{2} (2), 6, 3.

5. G is a subgroup of automorphisms of a surface of type V.

S_{4}, S_{3}.

6. G is a subgroup of automorphisms of a surface of type VI.

6, S_{3} × 2, S_{3}.

7. G is a subgroup of automorphisms of a surface of type VIII.

S_{3}.

6.6 Automorphisms of Del Pezzo surfaces of degree 2

Recall that the center of the Weyl group W (E(7)) is generated by an element w_{0} which acts on the root lattice as the negative of the identity. Its conjugacy class is of type A_{1}^{7}. The quotient group W (E_{7})′ = W (E_{7})/(⟨w_{0}⟩ is isomorphic to the simple group Sp(6, 𝔽_{2}). The extension 2.Sp(6, 𝔽_{2}) splits by the subgroup W (E_{7})^{+} equal to the kernel of the determinant homomorphism det : W (E_{7}) → {±1}. Thus we have

W (E_{7}) = W (E_{7})^{+} × ⟨w_{0}⟩.

Let H be a subgroup of W (E_{7})′. Denote by H^{+} its lift to an isomorphic subgroup of W^{+}. Any other isomorphic lift of H is defined by a nontrivial homomorphism α : H → ⟨w_{0}⟩ ≅ 2. Its elements are the products hα(h), h ∈ H^{+}. We denote such a lift by H_{α}. Thus all lifts are parametrized by the group Hom(H, ⟨w_{0}⟩)and H^{+} corresponds to the trivial homomorphism. Note that wH_{α}w^{−1} = (w′Hw′^{−1})_{α}, where w′ is the image of w in W (E_{7})′. In particular, two lifts of the same group are never conjugate.

Now we apply this to our geometric situation. Let S be a Del Pezzo surface of degree 2. Recall that the map S → ℙ^{2} defined by |−K_{S}| is a degree 2 cover. Its branch curve is a nonsingular curve of degree 4. It is convenient to view a Del Pezzo surface of degree 2 as a hypersurface in the weighted projective space ℙ(1, 1, 1, 2) given by an equation of degree 4

T_{3}^{2} + F_{4}(T_{0}, T_{1}, T_{2}) = 0.  (45)

The automorphism of the cover γ = [t_{0}, t_{1}, t_{2}, −t_{3}] defines the conjugacy class of a Geiser involution of ℙ^{2}. For any divisor class D on S we have D+γ_{0}^{∗}(D) ∈ | − mK_{S}| for some integer m. This easily implies that γ^{∗} acts as the minus identity in R_{S}. Its image in the Weyl group W (E_{7}) is the generator w_{0} of its center. Thus the Geiser involution is the geometric realization of w_{0}.

Let ρ : Aut(S) → W (E_{7}) be the natural injective homomorphism corresponding to a choice of a geometric basis in Pic(S). Denote by Aut(S)^{+} the full preimage of W (E_{7})^{+}. Since W (E_{7})^{+} is a normal subgroup, this definition is independent of a choice of a geometric basis. Under the restriction homomorphism Aut(S) → Aut(B) the group Aut(S)^{+} is mapped isomorphically to Aut(B) and we obtain

Aut(S)^{+} ≅ Aut(S)/⟨γ⟩ ≅ Aut(B).

¿From now on we will identify any subgroup G of Aut(B) with a subgroup of Aut(S) which we call the even lift of G. Under the homomorphism ρ : Aut(S) → W (E_{7}) all elements of G define even conjugacy classes, i.e. the conjugacy classes of elements from W (E_{7})^{+}. It is immediate to see that a conjugacy class is even if and only if the sum of the subscripts in its Carter graph is even. An isomorphic lift of a subgroup G to a subgroup of Aut(S) corresponding to some nontrivial homomorphism G → ⟨γ⟩ (or, equivalently to a subgroup of index 2 of G) will be called an odd lift of G.

The odd and even lifts of the same group are never conjugate, two minimal lifts are conjugate in Aut(S) if and only if the groups are conjugate in Aut(B). Two odd lifts of G are conjugate if and only if they correspond to conjugate subgroups of index 2 (inside of the normalizer of G in Aut(B)).

The following simple lemma will be heavily used.

Lemma 6.15. Let G be a subgroup of Aut(B) and H be its subgroup of index 2. Assume H is a minimal subgroup of Aut(S) (i.e. its even lift is such a subgroup). Then G is minimal in its even lift and its odd lift corresponding to H. Conversely, if G is minimal in both lifts, then H is a minimal subgroup.

Proof. Let Tr(g)_{±} be the trace of g ∈ G in the representation of G in R_{S} corresponding to the minimal (resp. odd) lift. Suppose G is minimal in both lifts. Then

∑_{g∈G} Tr_{+}(g) = ∑_{g∈H} Tr_{+}(g) + ∑_{g∉H} Tr_{+}(g) = 0,

∑_{g∈G} Tr_{−}(g) = ∑_{g∈H} Tr_{−}(g) + ∑_{g∉H} Tr_{−}(g) =

= ∑_{g∈H} Tr_{+}(g) − ∑_{g∉H} Tr_{+}(g) = 0.

This implies that ∑_{g∈H} Tr_{+}(g) = 0, i.e. H is minimal. The converse is obviously true.

Since γ generates a minimal subgroup of automorphisms of S, any group containing γ is minimal. So, we classify first subgroups of Aut(B) which admit minimal lifts. These will be all minimal subgroups of Aut(S) which do not contain the Geiser involution γ. The remaining minimal groups will be of the form ⟨γ⟩ × G̃, where G̃ is any lift of a subgroup G of Aut(B). Obviously, the product does not depend on the parity of the lift.

As in the case of cubic surfaces we first classify cyclic subgroups.

Lemma 6.16. Let g be an automorphism of order n > 1 of a nonsingular plane quartic C = V (F). Then one can choose coordinates in such a way that g = [t_{0}, ϵ_{n}^{a}t_{1}, ϵ_{n}^{b}t_{2}] and F is given in the following list.

(i) (n = 2), (a, b) = (0, 1),

F = T_{2}^{4} + T_{2}^{2}L_{2}(T_{0}, T_{1}) + L_{4}(T_{0}, T_{1}).

(ii)(n = 3), (a, b) = (0, 1),

F = T_{2}^{3}L_{1}(T_{0}, T_{1}) + L_{4}(T_{0}, T_{1}).

(iii)(n = 3), (a, b) = (1, 2),

F = T_{0}^{4} + αT_{0}^{2}T_{1}T_{2} + T_{0}T_{1}^{3} + T_{0}T_{2}^{3} + βT_{1}^{2}T_{2}^{2}.

(iv)(n = 4), (a, b) = (0, 1),

F = T_{2}^{4} + L_{4}(T_{0}, T_{1}).

(v) (n = 4), (a, b) = (1, 2),

F = T_{0}^{4} + T_{1}^{4} + T_{2}^{4} + αT_{0}^{2}T_{2}^{2} + βT_{0}T_{1}^{2}T_{2}.

(vi)(n = 6), (a, b) = (3, 2),

F = T_{0}^{4} + T_{1}^{4} + αT_{0}^{2}T_{1}^{2} + T_{0}T_{2}^{3}.

(vii)(n = 7), (a, b) = (3, 1),

F = T_{0}^{3}T_{1} + T_{1}^{3}T_{2} + T_{2}^{3}T_{0} + αT_{0}T_{1}^{2}T_{2}.

(viii) (n = 8), (a, b) = (3, 7),

F = T_{0}^{4} + T_{1}^{3}T_{2} + T_{1}T_{2}^{3}.

(ix)(n = 9), (a, b) = (3, 2),

F = T_{0}^{4} + T_{0}T_{1}^{3} + T_{2}^{3}T_{1}.

(x) (n = 12), (a, b) = (3, 4),

F = T_{0}^{4} + T_{1}^{4} + T_{0}T_{2}^{3}.

Here the subscript in polynomial L_{i} indicates its degree.

Also observe that the diagonal matrix (t, t, t, t^{2}) acts identically on S. Let g ∈ Aut(B) be an element of order n of type (∗) from the previous

Lemma. The following Table identifies the conjugacy class of two lifts g̃ of g in the Weyl group W (E_{7}). If n is even, then g admits two lifts in Aut(S) of order n. If n is odd, then one of the lifts is of order n and another is of order 2n. We denote by (∗)_{+} the conjugacy class of the lift which is represented by an element from W (E_{7})^{+} (of order n if n is odd). The conjugacy class of another lift is denoted by (∗)_{−}. The last column of the Table gives the trace of g on R_{S}.

The following is the list of elements of finite order which generate a minimal cyclic group of automorphisms. To identify the conjugacy class of a minimal lift we use the trace formula (36). If both lifts have the same trace, we distinguish them by computing the traces of their powers.

504  Igor V. Dolgachev and Vasily A. Iskovskikh
Type Order Notation Trace
(0)_{−}  2  7A_{1}  -7
(i)_{+}  2  4A_{1}  -1
(i)_{−}  2  3A_{1}  1
(ii)_{+}  3  3A_{2}  -2
(ii)_{−}  6  E_{7}(a_{4})  2
(iii)_{+}  3  2A_{2}  1
(iii)_{−}  6 D_{6}(a_{2}) + A_{1}  -1
(iv)_{+}  4  D_{4}(a_{1})  3
(iv)_{−}  4 2A_{3} + A_{1}  -3
(v)_{+}  4  2A_{3}  -1
(v)_{−}  4 D_{4}(a_{1}) + A_{1}  1
(vi)_{+}  6  E_{6}(a_{2})  2
(vi)_{−}  6  A_{2} + A_{5}  -2
(vii)_{+}  7  A_{6}  0
(vii)_{−} 14  E_{7}(a_{1})  0
(viii)_{+}  8  D_{5} + A_{1}  -1
(viii)_{−}  8  D_{5}  1
(ix)_{+}  9  E_{6}(a_{1})  1
(ix)_{−} 18  E_{7}  -1
(x)_{+} 12  E_{6}  0
(x)_{−} 12  E_{7}(a_{2})  0

Table 5. Conjugacy classes of automorphisms of a Del Pezzo surface of degree 2

1. Order 2 (A_{1}^{7}) (The Geiser involution) g = [t_{0}, t_{1}, t_{2}, −t_{3}] F = T_{3}^{2} + F_{4}(T_{0}, T_{1}, T_{2}).

2. Order 4 (2A_{3} + A_{1}) g = [t_{0}, t_{1}, it_{2}, t_{3}] F = T_{3}^{2} + T_{2}^{4} + L_{4}(T_{0}, T_{1}).

3. Order 6 (E_{7}(a_{4})) g = [t_{0}, t_{1}, ϵ_{3}t_{2}, −t_{3}] F = T_{3}^{2} + T_{2}^{3}L_{1}(T_{0}, T_{1}) + L_{4}(T_{0}, T_{1}).

4. Order 6 (A_{5} + A_{2}) g = [t_{0}, −t_{1}, ϵ_{3}t_{2}, −t_{3}] F = T_{3}^{2} + T_{0}^{4} + T_{1}^{4} + T_{0}T_{2}^{3} + aT_{0}^{2}T_{1}^{2}.

5. Order 6 (D_{6}(a_{2}) + A_{1}) g = [t_{0}, ϵ_{3}x_{1}, ϵ_{3}^{2}x_{2}, −x_{3}] F = T_{3}^{2} + T_{0}(T_{0}^{3} + T_{1}^{3} + T_{2}^{3}) + T_{1}T_{2}(αT_{0}^{2} + βT_{1}T_{2}).

6. Order 12 (E_{7}(a_{2})) g = [t_{0}, ϵ_{4}t_{1}, ϵ_{3}t_{2}, t_{3}] F = T_{3}^{2} + T_{0}^{4} + T_{1}^{4} + T_{0}T_{2}^{3}, (t_{0}, t_{1}, t_{2}, t_{3}).

7. Order 14 (E_{7}(a_{1})) g = [t_{0}, ϵ_{4}t_{1}, ϵ_{3}t_{2}, t_{3}]

F = T_{3}^{2} + T_{0}^{3}T_{1} + T_{1}^{3}T_{2} + T_{2}^{3}T_{0}.

8. Order 18 (E_{7}) g = [t_{0}, ϵ_{3}t_{1}, ϵ_{9}^{2}t_{2}, −t_{3}]

F = T_{3}^{2} + T_{0}^{4} + T_{0}T_{1}^{3} + T_{2}^{3}T_{1}.

Using the information about cyclic groups of automorphisms of plane quartics, it is not hard to get the classification of possible automorphism groups (see [26]). It is given in Table 5.

Type Order Structure  Equation  Parameters
I  336 2 × L_{2}(7)  T_{3}^{2} + T_{0}^{3}T_{1} + T_{1}^{3}T_{2} + T_{2}^{3}T_{0}
II  192 2 × (4^{2} : S_{3})  T_{3}^{2} + T_{0}^{4} + T_{1}^{4} + T_{2}^{4}
III  96  2 × 4A_{4}  T_{3}^{2} + T_{2}^{4} + T_{0}^{4} + aT_{0}^{2}T_{1}^{2} + T_{1}^{4}  a^{2} = −12
IV  48  2 × S_{4}  T_{3}^{2} + T_{2}^{4} + T_{1}^{4} + T_{0}^{4}+  a ≠ \frac{−1±\sqrt{−7}}{2}
+a(T_{0}^{2}T_{1}^{2} + T_{0}^{2}T_{2}^{2} + T_{1}^{2}T_{2}^{2})
V  32 2 × AS_{16}  T_{3}^{2} + T_{2}^{4} + T_{0}^{4} + aT_{0}^{2}T_{1}^{2} + T_{1}^{4} a^{2} ≠ 0, −12, 4, 36
VI  18  18  T_{3}^{2} + T_{0}^{4} + T_{0}T_{1}^{3} + T_{1}T_{2}^{3}
VII  16  2 × D_{8}  T_{3}^{2} + T_{2}^{4} + T_{0}^{4} + T_{1}^{4} + aT_{0}^{2}T_{1}^{2} + bT_{2}^{2}T_{0}T_{1}  a, b ≠ 0
VIII  12  2 × 6  T_{3}^{2} + T_{2}^{3}T_{0} + T_{0}^{4} + T_{1}^{4} + aT_{0}^{2}T_{1}^{2}
IX  12  2 × S_{3} T_{3}^{2} + T_{2}^{4} + aT_{2}^{2}T_{0}T_{1} + T_{2}(T_{0}^{3} + T_{1}^{3}) + bT_{0}^{2}T_{1}^{2}
X  8  2^{3}  T_{3}^{2} + T_{2}^{4} + T_{1}^{4} + T_{0}^{4} distinct a, b, c ≠ 0
+aT_{2}^{2}T_{0}^{2} + bT_{1}^{2}T_{2}^{2} + cT_{0}^{2}T_{1}^{2}
XI  6  6  T_{3}^{2} + T_{2}^{3}T_{0} + L_{4}(T_{0}, T_{1})
XII  4  2^{2}  T_{3}^{2} + T_{2}^{4} + T_{2}^{2}L_{2}(T_{0}, T_{1}) + L_{4}(T_{0}, T_{1})
XIII  2  2  T_{3}^{2} + F_{4}(T_{0}, T_{1}, T_{2})

Table 6. Groups of automorphisms of Del Pezzo surfaces of degree 2

Next we find minimal subgroups of automorphisms of a Del Pezzo surface of degree 2.

As in the previous case it is enough to consider surfaces S′ which are not specialized to surfaces S of other types. When this happens we write S′ → S. We have

IX → IV → I, II,
XII → X → V II → V → II, III,
XI → V III → III.

All of this is immediate to see, except the degeneration V III → III. This is achieved by some linear change of variables transforming the form x^{3}y + y^{4} to the form u^{4} + 2\sqrt{3}iu^{2}v^{2} + v^{4}. So it suffices to consider surfaces of types I, II, III, VI.

Before we start the classification we advice the reader to go back to the beginning of the section and recall the concepts of odd and even lifts of subgroups of Aut(B).

Type I.

Since L_{2}(7) has no subgroups of index 2 (in fact, it is a simple group), it admits a unique lift to a subgroup of Aut(S). It is known that the group L_{2}(7) is generated by elements of order 2, 3 and 7. Consulting Table 4, we find that an element of order 2 must be of type 4A_{1}, an element of order 3 must be of types 3A_{2} or 2A_{2}, and element of order 7 is of type A_{6}. To decide the type of a generator g of order 3, we use that it acts as a cyclic permutation of the coordinates in the plane, hence has 3 fixed points (1, 1, 1), (1, η_{3}, η_{3}^{2}), (1, η_{3}^{2}, η_{3}). The last two of them lie on the quartic. This easily implies that g has 4 fixed points on S, hence its trace in Pic(S) is equal to 2. This implies that g is of type 2A_{2}. Comparing the traces with the character table of the group L_{2}(7) we find that the representation of L_{2}(7) in (R_{S}) ⊗ ℂ is an irreducible 7-dimensional representation of L_{2}(7). Thus the group is minimal.

Assume G is a proper subgroup of L_{2}(7). It is known that maximal subgroups of L_{2}(7) are isomorphic to S_{4} or 7 : 3. There are two conjugacy classes of subgroups isomorphic to S_{4} (in the realization L_{2}(7) ≅ L_{3}(2) they occur as the stabilizer subgroups of a point or a line in ℙ^{2}(𝔽_{2})). Since S_{4} contains a unique subgroup of index 2, each subgroup can be lifted in two ways. Consider the even lift of S_{4} lying in L_{2}(7). To find the restriction of the 7-dimensional representation V_{7} = (R_{S})_{ℂ} to G we apply the Frobenius Reciprocity formula. Let χ_{k} denote a k-dimensional irreducible representation of L_{2}(7) and χ̄_{k} be its restriction to S_{4}. It is known that the induced character of the trivial representation of S_{4} is equal to χ_{1} + χ_{6} (see [18]). Applying the Frobenius Reciprocity formula we get ⟨χ̄_{1}, χ̄_{7}⟩ = ⟨χ_{1} + χ_{6}, χ_{7}⟩ = 0. This computation shows that the even lifts of the two conjugacy classes of S_{4} in L_{2}(7) are minimal subgroups. It follows from Lemma 6.15 the the odd lifts are minimal only if the lift of the subgroup A_{4} of S_{4} is minimal. One checks that the induced character of the trivial representation of A_{4} is equal to χ_{1} + χ_{6} + χ_{7}. By the Frobenius Reciprocity formula, the restriction of V_{7} to A_{4} contains the trivial summand. Thus A_{4} is not minimal and we conclude that there are only 2 non-conjugate lifts of S_{4} to a minimal subgroup of Aut(S).

Next consider the subgroup 7 : 3. It admits a unique lift. The induced representation of its trivial representation has the character equal to χ_{1} + χ_{7}. Applying the Frobenius Reciprocity formula, we see that this group is not minimal.

Let H be any subgroup of L_{2}(7) which admits a minimal lift. Since Aut(S) does not contain minimal elements of order 3 or 7, H must be a subgroup of S_{4}. Since A_{4} does not admit a minimal lift, H is either a cyclic group or isomorphic to either 2^{2} or D_{8}. The only cyclic group which may admit a minimal lift is a cyclic group of order 4. However, the character table for L_{2}(7) shows that the value of the character χ_{7} at an element of order 4 is equal to −1, hence it is of type 2A_{3}. It follows from the Table that this element does not admit minimal lifts.

Suppose G ≅ 2^{2}. In the even lift, it contains 3 nontrivial elements of type 4A_{1}. Adding up the traces we see that this group is not minimal. In the odd lift, it contains one element of type 4A_{1} and two of type 3A_{1}. Again, we see that the group is not minimal.

Assume G ≅ D_{8}. The group S_{4} is the normalizer of D_{8}. This shows that there are two conjugacy classes of subgroups isomorphic to D_{8}. The group G admits 2 lifts. In the even lift it contains 2 elements of type 2A_{3} and 5 elements of type 4A_{1}. Adding up the traces, we obtain that the lift is minimal. Since the lift of 4 is not minimal, the odd lift of D_{8} is not minimal.

Type II.

The group Aut(B) is generated by the transformations

g_{1} = [t_{0}, it_{1}, t_{2}, −t_{3}], τ = [t_{1}, t_{0}, t_{2}, t_{3}], σ = [t_{0}, t_{2}, t_{1}, t_{3}]

of types D_{4}(a_{1}), 4A_{1}, 4A_{1}, respectively. Let g_{2} = σg_{1}σ^{−1} = [t_{0}, t_{1}, it_{2}, −t_{3}].

We have

τg_{1}τ^{−1} = g_{1}^{−1}g_{2}^{−1}, τg_{2}τ^{−1} = g_{2}.

The elements g_{1}, g_{2}, γ generate a normal subgroup isomorphic to 4^{2}. The quotient group is isomorphic to S_{3}. Its generators of order 2 can be represented by τ and σ. The elements g_{1}^{2}, g_{2}^{2}, τ, σ generate a subgroup (not normal) isomorphic to S_{4}. Thus

Aut(B) ≅ 4^{2} : S_{3}  (46)

and

Aut(S) ≅ 2 × (4^{2} : S_{3}).  (47)

Consider the natural homomorphism f : Aut(B) → S_{3} with kernel 4^{2}. We will consider different cases corresponding to a possible image of a subgroup G ⊂ Aut(B) in S_{3}. For the future use we observe that Aut(B) does not contain elements of order 6 because its square is an element of type 3A_{2} but all our elements of order 3 are of type 2A_{2}. Also it does not contain 2^{3} (this follow from the presentation of the group). We will also use that Aut(B) contains 2 conjugacy classes of elements of order 4 of types D_{4}(a_{1}) (represented by g_{1}) and 2A_{3} (represented by g_{1}g_{2}).

Case 3 : f(G) = {1}.

In this case G is a subgroup of 4^{2}. The group itself contains 3 elements of type 4A_{1}, 6 elements of type D_{4}(a_{1}) and 6 elements of type 2A_{3}. The sum of the traces is equal to 16. Thus the group is not minimal. So no subgroup is minimal in the even lift. An odd lift corresponding to the homomorphism 4^{2} → ⟨γ⟩ sending an element of type D_{4}(a_{1}) to γ defines an odd lift. There is only one conjugacy class of subgroups of index 2 in 4^{2}. It defines an odd lift of 4^{2}. We may assume that the subgroup of index 2 is generated by g_{1}, g_{2}^{2}. It admits two odd lifts corresponding to the subgroups ⟨g_{1}^{2}, g_{2}^{2}⟩ and ⟨g_{1}g_{2}^{2}⟩. Finally a cyclic subgroup 4 of type D_{4}(a_{1}) admits an odd lift. No other subgroup admits a minimal lift.

Case 2 : |f(G)| = 2.

Replacing the group by a conjugate group, we may assume that f(G) = ⟨τ⟩. We have

G_{1} = f^{−1}(⟨τ⟩) = ⟨τ, g_{1}, g_{2}⟩ ≅ 4^{2} : 2 ≅ 4D_{8},

where the center is generated by g_{2}.

Let H = ⟨τ, g_{1}^{2}g_{2}⟩. One immediately checks that H is normal in G_{1} and isomorphic to D_{8}. We have G_{1} ≅ D_{8} : 4. The subgroup H consists of 5 elements of type 4A_{1} and 2 elements of type 2A_{3}. Adding up the traces we obtain that H is minimal in its even lift. Thus G_{1} is minimal in its even lift. The subgroup G_{2} = ⟨τ, g_{1}^{2}, g_{2}⟩ is of order 16. It contains H defining a split extension D_{8} : 2 with center generated by g_{2}. It is isomorphic to the group AS_{16} (see Table 1) is of index 2 in G_{1}. Since it is minimal, the odd lift of G_{1} corresponding to this subgroup is minimal.

We check that τg_{1} is of order 8 and the normalizer of the cyclic group ⟨τg_{1}⟩ is generated by this subgroup and g_{1}^{2}. This gives us another subgroup G_{3} of index 2 of G_{1}. It is a group of order 16 isomorphic to M_{16}. An element of order 8 is of type D_{5}(a_{1}) + A_{1}. Thus the sum of the traces is equal to 8. Adding up the traces of elements in the nontrivial coset of ⟨τg_{1}⟩ we get that the sum is equal to −8 (all elements have the trace equal to −1). This shows that G_{3} is minimal. Thus the corresponding odd lift of G_{1} is minimal.

Let G be a subgroup of index 2 of G_{1} and g = τg_{1}^{a}g_{2}^{b} ∈ G be the element of largest possible order in H. We verify that g^{2} = g_{2}^{2b−a}. If g is of order 8, we check that it generates either ⟨τg_{1}⟩ considered earlier or its conjugate subgroup. Its normalizer is conjugate to the subgroup G_{3} considered earlier. If g is of order 4, then 2b − a ≡ 2 mod 4. We list all possible cases and find that all elements of order 4 are conjugate. Thus we may assume that G contains g = τg_{1}^{2}. Now we check that the normalizer of this group is our group G_{2}.

So, all subgroups of index 2 are accounted for. They are two of them isomorphic to AS_{16} and M_{16}. They are all minimal in their even lift, and hence define odd lifts of G_{1}.

Let G be a subgroup of index 4 of f^{−1}(τ). It follows from above argument that G is conjugate to a subgroup of index 2 of G_{2} or G_{3}. It could be D_{8}, 8, or 2 × 4. The first group is minimal, hence D_{8} : 2 admits an odd minimal lift. Other two groups are not minimal. The last group admits an odd minimal lift. Note that it is not conjugate to odd 2 × 4 from Case 1. Finally a cyclic group of type D_{4}(a_{1}) admits an odd minimal lift. It is not conjugate to a group from Case 1.

Case 3 : |f(G)| = 3.

Without loss of generality we may assume that f(G) = ⟨στ⟩. By Lemma 4.2, G is a split extension H : 3, where H is a subgroup of ⟨g_{1}, g_{2}⟩. Let G_{1} = f^{−1}(⟨στ⟩). It is a split extension 4^{2} : 3. By Sylow’s Theorem, all subgroups of order 3 are conjugate. Thus we may assume that H contains στ. The possibilities are G_{1} or G_{2} = ⟨g_{1}^{2}, g_{2}^{2}, στ⟩ ≅ 2^{2} : 3 ≅ A_{4}. The group A_{4} has 3 elements of type 4A_{1}, 4 elements of type 2A_{2} and 4 elements of type 3A_{2}. Adding up the traces we see that the group is minimal. Thus G_{1} is minimal too. The group G_{1} does not have subgroups of index 2, so it does not admit odd lifts. Other groups in this case are conjugate to the non-minimal group ⟨στ⟩.

Case 4 :f(G) = S_{3}.

In this case G ∩ f^{−1}(⟨στ⟩) is a subgroup of index 2 equal to one of the two groups considered in the previous case. We get G = Aut(B), or G ≅ 2^{2} : S_{3} ≅ S_{4}, or S_{3}. Considering the preimage of ⟨τ⟩, we find that all groups isomorphic to S_{4} are conjugate and their Sylow 2-subgroup is D_{8} from the previous case. Thus both Aut(B) and S_{4} admit two minimal lifts. A group isomorphic to S_{3} contains 2 elements of type 2A_{2} and it is not minimal in any lift.

Type III.

We assume that a = 2\sqrt{3}i in the equation of the surface. The group Aut(B) is isomorphic to 4A_{4}. It is generated (as always in its even lift) by

g_{1} = [t_{1}, t_{0}, t_{2}, −t_{3}], g_{2} = [it_{1}, −it_{0}, t_{2}, −t_{3}],
g_{3} = [ϵ_{8}^{7}t_{0} + ϵ_{8}^{7}t_{1}, ϵ_{8}^{5}t_{0} + ϵ_{8}t_{1}, \sqrt{2}ϵ_{12}t_{2}, 2ϵ_{6}t_{3}], c = [t_{0}, t_{1}, it_{2}, −t_{3}]

The “complicated” transformation g_{3} is of order 3 (see our list of Gründformen for binary polyhedral groups). The generators g_{1}, g_{2} are of type 4A_{1}, the generator g_{3} is of type 2A_{2} and the generator c is of type D_{4}(a_{1}).

The element c generates the center. We have g_{1}g_{2} = g_{2}g_{1}c^{2}. This shows that the quotient by ⟨c⟩ is isomorphic to A_{4} and the subgroup ⟨c, g_{1}, g_{2}⟩ ≅ 4D_{4} is a group of order 16 isomorphic to the group AS_{16} (see Table 1).

Let f : Aut(B) → A_{4} be the natural surjection with kernel ⟨c⟩. Let G be a subgroup of Aut(B).

Case 1 : G ⊂ Ker(f) ≅ 4.

There are no even minimal subgroups. The whole kernel admits a minimal odd lift.

Case 2 : #f(G) = 2.

Without loss of generality we may assume that f(G) = ⟨g_{1}⟩. The subgroup f^{−1}(⟨g_{1}⟩) is generated by c, g_{1} and is isomorphic to 4 × 2. It is not minimal in the even lift and minimal in the unique odd lift. Its subgroup ⟨cg_{1}⟩ of order 4 is of type 2A_{3} and does not admit minimal lifts.

Case 3 :f(G) = ⟨g_{3}⟩.

We have G = f^{−1}(⟨g_{3}⟩) = ⟨c, g_{3}⟩ = ⟨cg_{3}⟩ ≅ 12. The element cg_{3} is of type E_{6}, hence not minimal. Its square is an element of type E_{6}(a_{2}), also not minimal. The subgroup ⟨(cg_{3})^{2}⟩ ≅ 6 defines an odd minimal lift of G. The subgroup ⟨(cg_{3})^{4}⟩ is of order 3. It defines an odd minimal lift of ⟨(cg_{3})^{2}⟩. The group ⟨g_{3}⟩ admits an odd minimal lift.

Case 4 : f(G) = ⟨g_{1}, g_{2}⟩ ≅ 2^{2}.

The subgroup H = f^{−1}(⟨g_{1}, g_{2}⟩) is generated by c, g_{1}, g_{2}. As we observed earlier, it is isomorphic to the group AS_{16} from Table 1. A proper subgroup is conjugate in Aut(B) to either ⟨g_{1}, g_{2}⟩ ≅ D_{8} or ⟨cg_{1}, g_{2}⟩. All of the subgroups are isomorphic to D_{8} with center generated by c^{2}. The cyclic subgroup of order 4 is of type 2A_{3}, thus the subgroups are minimal in the even lift (we have done this computation for surfaces of type II). Thus the group H is minimal in the even lift and also minimal in two odd lifts corresponding to its two subgroups of index 2.

Case 5 : f(G) = A_{4}.

It is easy to see that G has non-trivial center (the center of its Sylow 2subgroup). It is equal to ⟨c⟩ or ⟨c^{2}⟩. In the first case G = Aut(B). Since it contains minimal subgroups it is minimal.

A subgroup G of index 2 is isomorphic to 2A_{4} ≅ D_{8} : 3. Its Sylow 2subgroup is equal to one of the two subgroups isomorphic to D_{8} from Case 4. Thus Aut(B) admits two odd lifts. Since G has no subgroups of index 2, the odd lifts of G do not exist.

Type VI.

In this case Aut(B) ≅ 9 is not minimal so does not admit minimal lifts. To summarize our investigation we give two lists. In the first we list all

groups which do not contain the Geiser involution γ. We indicate by + or − the types of their lifts. Also we indicate the number of conjugacy classes.

All other minimal groups are of the form ⟨γ⟩ × G, where G is one of the lifts of a subgroup of Aut(B). In the second list we give only groups 2 × G, where G does not admit a minimal lift. All other groups are of the form 2×G, where G is given in the previous table.

Theorem 6.17. Let G be a minimal group of automorphisms of a Del Pezzo surface of degree 2. Then G is either equal to a minimal lift of a subgroup from Table 7 or equal to γ × G′, where G′ is either from the table of one of the following groups of automorphisms of the branch quartic curve B

1. Type I: 7 : 3, A_{4}, S_{3}, 7, 4, 3, 2.

2. Type II: 2^{2}, S_{3}, 8, 4, 3, 2.

3. Type III: 2^{2}, 4, 2.

4. Type IV: S_{3}, 2^{2}, 3, 2.

5. Type V: 2^{2}, 2.

6. Type VI: 9, 3.

7. Type VII: 2^{2}, 4, 2

8. Type VIII: 3.

9. Type IX: S_{3}, 3, 2.

10. Type X: 2^{2}, 2.

11. Type XI: 3.

12. Type XII: {1}.

Type of S  Group Lift Conjugacy classes
I L_{2}(7) + 1
S_{4} +  2
D_{8} +  2
II 4^{2} : S_{3} +,- 2
S_{4} +,-  2
4^{2} : 3 +  1
A_{4} +  1
4^{2} : 2 ≅ D_{8} : 4 +,-,-  3
M_{16} +  1
AS_{16} +,-  2
D_{8} +  1
4^{2}  -  1
2 × 4  -  2
4  -  2
III  4A_{4} +,-  2
2A_{4} ≅ D_{8} : 3 +  1
AS_{16} +,-  2
D_{8} +  1
12  -  1
6  -  1
2 × 4  -  1
4  -  1
IV  S_{4} +  1
D_{8} +  1
V AS_{16} +,- 2
D_{8} +  1
2 × 4  -  2
4  -  1
VII  D_{8} +  1
VIII  6  -  1

Table 7. Minimal groups of automorphisms not containing γ

6.7 Automorphisms of Del Pezzo surfaces of degree 1

Let S be a Del Pezzo surface of degree 1. The linear system | − 2K_{S}| defines a finite map of degree 2 onto a quadric cone Q in ℙ^{3}. Its branch locus is a nonsingular curve B of genus 4 cut out by a cubic surface. Recall that a singular quadric is isomorphic to the weighted projective space ℙ(1, 1, 2). A curve of genus 4 of degree 6 cut out in Q by a cubic surface is given by equation F (T_{0}, T_{1}, T_{2}) of degree 6. After change of coordinates it can be given by an equation T_{2}^{3} + F_{4}(T_{0}, T_{1})T_{2} + F_{6}(T_{0}, T_{1}) = 0, where F_{4} and F_{6} are binary forms of degree 4 and 6. The double cover of Q branched along such curve is isomorphic to a hypersurface of degree 6 in ℙ(1, 1, 2, 3)

T_{3}^{2} + T_{2}^{3} + F_{4}(T_{0}, T_{1})T_{2} + F_{6}(T_{0}, T_{1}) = 0.  (48)

The vertex of Q has coordinates (0, 0, 1) and its pre-image in the cover consist of one point (0, 0, 1, a), where a^{2} + 1 = 0 (note that (0, 0, 1, a) and (0, 0, 1, −a) represent the same point on ℙ(1, 1, 2, 3)). This is the base-point of | − K_{S}|. The members of | − K_{S}| are isomorphic to genus 1 curves with equations y^{2} + x^{3} + F_{4}(t_{0}, t_{1})x + F_{6}(t_{0}, t_{1}) = 0. The locus of zeros of Δ = F_{4}^{3} + 27F_{6}^{2} is the set of points in ℙ^{1} such that the corresponding genus 1 curve is singular. It consists of a simple roots and b double roots. The zeros of F_{4} are either common zeros with F_{6} and Δ, or represent nonsingular elliptic curves isomorphic to an anharmonic plane cubic curve. The zeros of F_{6} are either common zeros with F_{4} and Δ, or represent nonsingular elliptic curves isomorphic to a harmonic plane cubic curve.

Observe that no common root of F_{4} and F_{6} is a multiple root of F_{6} since otherwise the surface is singular.

Since the ramification curve of the cover S → Q (identified with the branch curve B) is obviously invariant with respect to Aut(S) we have a natural surjective homomorphism

Aut(S) → Aut(B).  (49)

Its kernel is generated by the deck involution β which we call the Bertini involution. It defines the Bertini involution in Cr(2). The Bertini involution is the analog of the Geiser involution for Del Pezzo surfaces of degree 2. The same argument as above shows that β acts in R_{S} as the minus of the identity map. Under the homomorphism Aut(S) → W (E_{8}) defined by a choice of a geometric basis, the image of β is the elements w_{0} generating the center of W (E_{8}). This time w_{0} is an even element, i.e. belongs to W (E_{8})^{+}. The quotient group W (E_{8})^{+}/⟨w_{0}⟩ is isomorphic to the simple group O(8, 𝔽_{2})^{+}.

Since Q is a unique quadric cone containing B, the group Aut(B) is a subgroup of Aut(Q). Consider the natural homomorphism

r : Aut(B) → Aut(ℙ^{1}).

Let G be a subgroup of Aut(B) and P be its image in Aut(ℙ^{1}). We assume that elements from G act on the variables T_{0}, T_{1} by linear transformations with determinant 1. The polynomials F_{4} and F_{6} are the relative invariants of the binary group P̄. They are polynomials in Gründformen which were listed in section 5.5. Let χ_{4}, χ_{6} be the corresponding characters of P̄ defined by the binary forms F_{4}, F_{6}. Let χ_{2}, χ_{3} be the characters of G defined by the action on the variables T_{2}, T_{3}. Assume that F_{4} ≠ 0. Then

χ_{4}χ_{2} = χ_{6} = χ_{3}^{3} = χ_{3}^{2}.

If g ∈ G∩Ker(r)\{1}, then g acts on the variables T_{0}, T_{1} by either the identity or the minus identity. Thus χ_{4}(g) = χ_{6}(g) = 1 and we must have χ_{2}(g) = χ_{3}(g)^{2} = 1. This shows that g = [t_{0}, t_{1}, t_{3}, −t_{3}] = [−t_{0}, −t_{1}, t_{2}, −t_{3}] = β.

If F_{4} = 0, then we must have only χ_{2}(g)^{3} = χ_{3}(g)^{2} = 1. Since [−t_{0}, −t_{1}, t_{2}, −t_{3}] is the identity transformation, we may assume that χ_{3}(g) = 1 and represent g by g = [t_{0}, t_{1}, ϵ_{3}t_{3}, ±t_{3}]. Thus G ∩ Ker(r) = ⟨β, α⟩ ≅ 6.

Conversely, start with a polyhedral group P such that its lift to a binary polyhedral group P̄ acts on the variables T_{0}, T_{1} leaving V (F_{4}) and V (F_{6}) invariant. Let χ_{4}, χ_{6} be the corresponding characters. Assume that there exist character χ_{2}, χ_{3} : P̄ → ℂ^{∗} such that

χ_{0}^{2} = χ_{4}χ_{1} = χ_{6} = χ_{1}^{3}.  (50)

Then g = [at_{0}+bt_{1}, ct_{0}+t_{1}] ∈ P̄ acts on S by [at_{0}+bt_{1}, ct_{0}+t_{1}, χ_{2}(g)t_{2}, χ_{3}(t_{3})]. This transformation is the identity in Aut(S) if and only if g = [−t_{0}, −t_{1}] and χ_{2}(−1) = 1, χ_{3}(−1) = −1. This shows that P̄ can be identified with a subgroup of Aut(S) with −I_{2} = β if and only if χ_{3}(−1) = −1. If χ_{3}(−1) = 1, then P can be identified with a subgroup of Aut(S) not containing β. In the latter case,

r^{−1}(P) = \begin{cases}P × ⟨β⟩ & if F_{4} ≠ 0 \\ P × ⟨β, α⟩ & otherwise. \end{cases}

In particular, if F_{4} = 0, there are three subgroups of Aut(S) which are mapped surjectively to P.

In the former case

r^{−1}(P) = \begin{cases}P̄ & if F_{4} ≠ 0, \\ P̄ × ⟨α⟩ & otherwise. \end{cases}

Of course it could happen that neither P nor P̄ lifts to a subgroup of Aut(S). In this case r^{−1}(P) ≅ 2P ≅ ¯P̸ or r^{−1}(P) ≅ 3 × 2P (if F_{4} = 0).

In the following list we give a nontrivial subgroup P of Aut(ℙ^{1}) as a group of automorphisms of B and a smallest lift P̃ of P to a subgroup of r^{−1}(P). If F_{4} ≠ 0, then we will see that P̃ = r^{−1}(P) or P̃ ≅ P. In the latter case r^{−1}(P) ≅ 2 × P. If F_{4} = 0, and P̃ ≅ P, then r^{−1}(P) ≅ 6 × P̃. Otherwise r^{−1}(P) ≅ 3 × P̃.

Also we give generators of P̃ to Aut(S) as a group acting on t_{0}, t_{1} with determinant 1 and the Bertini involution as an element of the lift.

1. Cyclic groups P (i) P = {1}, F_{4} = 0, r^{−1}(P) = ⟨β, α⟩ ≅ 6. (ii) P ≅ 2, g = [it_{0}, −it_{1}, −t_{2}, it_{3}],

F_{4} = F_{2}(T_{0}^{2}, T_{1}^{2}) ≠ 0, F_{6} = F_{3}(T_{0}^{2}, T_{1}^{2});

(iii)P ≅ 2, P̃ ≅ 4, g = [it_{0}, −it_{1}, t_{2}, t_{3}], β = g^{2},

F_{4} = a(T_{0}^{4} + T_{1}^{4}) + bT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}T_{1}F_{2}(T_{0}^{2}, T_{1}^{2});

(iv)P ≅ 2, P̃ = 4, g = [−t_{1}, t_{0}, t_{2}, t_{3}], β = g^{2},

F_{4} as in (iii), F_{6} = (T_{0}^{2} + T_{1}^{2})(a(T_{0}^{4} + T_{1}^{4}) + T_{0}T_{1}(bT_{0}T_{1} + c(T_{0}^{2} − T_{1}^{2}));

(v) P ≅ 2, g = [−t_{1}, t_{0}, −t_{2}, it_{3}],

F_{4} as in (iii), F_{6} = a(T_{0}^{6} − T_{1}^{6}) + bT_{0}T_{1}(T_{0}^{4} + T_{1}^{4});

(vi)P ≅ 3, g = [ϵ_{3}t_{0}, ϵ_{3}^{2}t_{1}, ϵ_{3}^{2}t_{2}, t_{3}],

F_{4} = T_{0}(aT_{0}^{3} + bT_{1}^{3}), F_{6} = F_{2}(T_{0}^{3}, T_{1}^{3});

(vii)P ≅ 3, g = [ϵ_{3}t_{0}, ϵ_{3}^{2}t_{1}, t_{2}, t_{3}],

F_{4} = aT_{0}^{2}T_{1}^{2}, F_{6} = F_{2}(T_{0}^{3}, T_{1}^{3});

(viii) P ≅ 4, g = [ϵ_{8}t_{0}, ϵ_{8}^{−1}t_{1}, it_{2}, ϵ_{8}^{3}t_{3}],

F_{4} = aT_{0}^{4} + bT_{1}^{4}, F_{6} = T_{0}^{2}(cT_{0}^{4} + dT_{1}^{4});

(ix)P ≅ 4, P̃ ≅ 8, g = [ϵ_{8}t_{0}, ϵ_{8}^{−1}t_{1}, −t_{2}, t_{3}], β = g^{4},

F_{4} = aT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}T_{1}(T_{0}^{4} + T_{1}^{4});

(x) P ≅ 5, g = [ϵ_{10}t_{0}, ϵ_{10}^{−1}t_{1}, ϵ_{5}t_{2}, ϵ_{10}^{3}t_{3}],

F_{4} = aT_{0}^{4}, F_{6} = T_{0}(T_{0}^{5} + T_{1}^{5});

(xi)P ≅ 6, g = [ϵ_{12}t_{0}, ϵ_{12}^{−1}t_{1}, ϵ_{6}t_{2}, it_{3}],

F_{4} = aT_{0}^{4}, F_{6} = bT_{0}^{6} + T_{1}^{6}, b ≠ 0;

(xii)P ≅ 6, g = [ϵ_{12}t_{0}, ϵ_{12}^{−1}t_{1}, −t_{2}, it_{3}],

F_{4} = aT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}^{6} + T_{1}^{6};

(xiii) P ≅ 10, P̃ ≅ 20, g = [ϵ_{20}t_{0}, ϵ_{20}^{−1}t_{1}, ϵ_{10}^{8}t_{2}, ϵ_{10}^{−1}t_{3}], g^{10} = β,

F_{4} = T_{0}^{4}, F_{6} = T_{0}T_{1}^{5};

(xiv) P ≅ 12, g = [ϵ_{24}t_{0}, ϵ_{24}^{−1}t_{1}, ϵ_{12}t_{2}, ϵ_{24}t_{3}],

F_{4} = T_{0}^{4}, F_{6} = T_{1}^{6}.

2. Dihedral groups

(i) P ≅ 2^{2}, P̃ ≅ D_{8}, g_{1} = [it_{1}, it_{0}, −t_{2}, it_{3}], g_{2} = [−t_{1}, t_{0}, −t_{2}, it_{3}], β = (g_{1}g_{2})^{2}, g_{1}^{2} = g_{2}^{2} = 1,

F_{4} = a(T_{0}^{4} + T_{1}^{4}) + bT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}T_{1}[c(T_{0}^{4} + T_{1}^{4}) + dT_{0}^{2}T_{1}^{2}];

(ii) P ≅ 2^{2}, P̃ ≅ Q_{8}, g_{1} = [it_{1}, it_{0}, t_{2}, t_{3}], g_{2} = [−t_{1}, t_{0}, t_{2}, t_{3}], β = g_{1}^{2} = g_{2}^{2},

F_{4} = a(T_{0}^{4} + T_{1}^{4}) + bT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}T_{1}(T_{0}^{4} − T_{1}^{4});

(iii)P ≅ D_{6}, g_{1} = [ϵ_{6}t_{0}, ϵ_{6}^{−1}t_{1}, t_{2}, −t_{3}], g_{2} = [it_{1}, it_{0}, −t_{2}, it_{3}],

F_{4} = aT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}^{6} + T_{1}^{6} + bT_{0}^{3}T_{1}^{3};

(iv)P ≅ D_{8}, P̃ ≅ D_{16}, g_{1} = [ϵ_{8}t_{0}, ϵ_{8}^{−1}t_{1}, −t_{2}, it_{3}], g_{2} = [−t_{1}, t_{0}, −t_{2}, it_{3}], g_{1}^{4} = β, g_{2}^{2} = 1,

F_{4} = aT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}T_{1}(T_{0}^{4} + T_{1}^{4});

(v) P ≅ D_{12}, P̃ ≅ 2D_{12} ≅ (2 × 6)^{•}2, g_{1} = [ϵ_{12}t_{0}, ϵ_{12}^{−1}t_{1}, −t_{2}, it_{3}], g_{2} = [−t_{1}, t_{0}, t_{2}, t_{3}], g_{1}^{6} = 1, g_{2}^{2} = β,

F_{4} = aT_{0}^{2}T_{1}^{2}, F_{6} = T_{0}^{6} + T_{1}^{6}.

3. Other groups

(i) P ≅ A_{4}, P̃ ≅ \overline{T} , g_{1} = [ϵ_{8}^{7}t_{0} + ϵ_{8}^{7}t_{1}, ϵ_{8}^{5}t_{0} + ϵ_{8}t_{1}, \sqrt{2}ϵ_{3}t_{2}, 2t_{3}], g_{2} = [it_{0}, −it_{1}, t_{2}, t_{3}], g_{1}^{3} = g_{2}^{2} = β,

F_{4} = T_{0}^{4} + 2\sqrt{−3}T_{0}^{2}T_{1}^{2} + T_{2}^{4}, F_{6} = T_{0}T_{1}(T_{0}^{4} − T_{1}^{4}),

(ii) P ≅ O, P̃ ≅ \overline{T} : 2, g_{1} = [ϵ_{8}^{7}t_{0} + ϵ_{8}^{7}t_{1}, ϵ_{8}^{5}t_{0} + ϵ_{8}t_{1}, \sqrt{2}ϵ_{3}t_{2}, 2t_{3}], g_{2} = [ϵ_{8}t_{0}, ϵ_{8}^{−1}t_{1}, −t_{2}, it_{3}], g_{3} = [−ϵ_{8}t_{1}, ϵ_{8}^{7}t_{0}, −t_{2}, it_{3}], g_{1}^{3} = g_{2}^{4} = β, g_{3}^{2} = 1, r^{−1}(P) = 3 × Ō,

F_{4} = 0, F_{6} = T_{0}T_{1}(T_{0}^{4} − T_{1}^{4}),

Table 6 gives the list of the full automorphism groups of Del Pezzo surfaces of degree 1.

The following is the list of cyclic minimal groups ⟨g⟩ of automorphisms of Del Pezzo surfaces V (F) of degree 1.

1. Order 2 • A_{1}^{8} (the Bertini involution) g = [t_{0}, t_{1}, t_{2}, −t_{3}]

F = T_{3}^{2} + T_{2}^{3} + F_{4}(T_{0}, T_{1})T_{2} + F_{6}(T_{0}, T_{1}),

2. Order 3 • 4A_{2} g = [t_{0}, t_{1}, ϵ_{3}t_{2}, t_{3}]

F = T_{3}^{2} + T_{2}^{3} + F_{6}(T_{0}, T_{1}),

3. Order 4 • 2D_{4}(a_{1}) g = [t_{0}, −t_{1}, −t_{2}, ±it_{3}]

F = T_{3}^{2} + T_{2}^{3} + (aT_{0}^{4} + bT_{0}^{2}T_{1}^{2} + cT_{1}^{4})T_{2} + T_{0}T_{1}(dT_{0}^{4} + eT_{1}^{4}),

4. Order 5

516  Igor V. Dolgachev and Vasily A. Iskovskikh
Type Order Structure  F_{4}  F_{6} Parameters
I  144 3 × (\overline{T} : 2)  0  T_{0}T_{1}(T_{0}^{4} − T_{1}^{4})
II  72 3 × 2D_{12}  0  T_{0}^{6} + T_{1}^{6}
III  36 6 × D_{6}  0  T_{0}^{6} + aT_{0}^{3}T_{1}^{3} + T_{1}^{6}  a ≠ 0
IV  30  30  0  T_{0}(T_{0}^{5} + T_{1}^{5})
V  24  \overline{T} a(T_{0}^{4} + αT_{0}^{2}T_{1}^{2} + T_{1}^{4})  T_{0}T_{1}(T_{0}^{4} − T_{1}^{4}) α = 2\sqrt{−3}
VI  24  2D_{12}  aT_{0}^{2}T_{1}^{2}  T_{0}^{6} + T_{1}^{6}  a ≠ 0
VII  24  2 × 12  T_{0}^{4}  T_{1}^{6}
VIII  20  20  T_{0}^{4}  T_{0}T_{1}^{5}
IX  16  D_{16}  aT_{0}^{2}T_{1}^{2}  T_{0}T_{1}(T_{0}^{4} + T_{1}^{4})  a ≠ 0
X  12  D_{12}  T_{0}^{2}T_{1}^{2}  T_{0}^{6} + aT_{0}^{3}T_{1}^{3} + T_{1}^{6}  a ≠ 0
XI  12  2 × 6  0  G_{3}(T_{0}^{2}, T_{1}^{2})
XII  12  2 × 6  T_{0}^{4}  aT_{0}^{6} + T_{1}^{6}  a ≠ 0
XIII  10  10  T_{0}^{4}  T_{0}(aT_{0}^{5} + T_{1}^{5})  a ≠ 0
XIV  8  Q_{8} T_{0}^{4} + T_{1}^{4} + aT_{0}^{2}T_{1}^{2}  bT_{0}T_{1}(T_{0}^{4} − T_{1}^{4}) a ≠ 2\sqrt{−3}
XV  8  2 × 4  aT_{0}^{4} + T_{1}^{4}  T_{0}^{2}(bT_{0}^{4} + cT_{1}^{4})
XVI  8  D_{8} T_{0}^{4} + T_{1}^{4} + aT_{0}^{2}T_{1}^{2} T_{0}T_{1}(b(T_{0}^{4} + T_{1}^{4}) + cT_{0}^{2}T_{1}^{2})  b ≠ 0
XVII  6  6  0  F_{6}(T_{0}, T_{1})
XVIII  6  6  T_{0}(aT_{0}^{3} + bT_{1}^{3})  cT_{0}^{6} + dT_{0}^{3}T_{1}^{3} + T_{1}^{6}
XIX  4  4  G_{2}(T_{0}^{2}, T_{1}^{2}  T_{0}T_{1}F_{2}(T_{0}^{2}, T_{1}^{2})
XX  4  2^{2}  G_{2}(T_{0}^{2}, T_{1}^{2}  G_{3}(T_{0}^{2}, T_{1}^{2})
XXI  2  2  F_{4}(T_{0}, T_{1})  F_{6}(T_{0}, T_{1})

Table 8. Groups of automorphisms of Del Pezzo surfaces of degree 1

• 2A_{4} g = [t_{0}, ϵ_{5}t_{1}, t_{2}, t_{3}]

F = T_{3}^{2} + T_{2}^{3} + aT_{0}^{4}T_{2} + T_{0}(bT_{0}^{5} + T_{1}^{5}),

5. Order 6

• E_{6}(a_{2}) + A_{2} g = [t_{0}, −t_{1}, ϵ_{3}t_{2}, t_{3}] F = T_{3}^{2} + T_{2}^{3} + G_{3}(T_{0}^{2}, T_{1}^{2}),

• E_{7}(a_{4}) + A_{1} g = [t_{0}, ϵ_{3}t_{1}, t_{2}, −t_{3}] F = T_{3}^{2} + T_{2}^{3} + (T_{0}^{4} + aT_{0}T_{1}^{3})T_{2} + bT_{0}^{6} + cT_{0}^{3}T_{1}^{3} + dT_{1}^{6},

• 2D_{4} g = [ϵ_{6}t_{0}, ϵ_{6}^{−1}t_{1}, t_{2}, t_{3}] F = T_{3}^{2} + aT_{0}^{2}T_{1}^{2}T_{2} + bT_{0}^{6} + cT_{0}^{3}T_{1}^{3} + eT_{1}^{6},

• E_{8}(a_{8}) g = [t_{0}, t_{1}, ϵ_{3}t_{2}, −t_{3}] F = T_{3}^{2} + T_{2}^{3} + F_{6}(T_{0}, T_{1}),

• A_{5} + A_{2} + A_{1} g = [t_{0}, ϵ_{6}t_{1}, t_{2}, t_{3}] F = T_{3}^{2} + T_{2}^{3} + aT_{0}^{4}T_{2} + T_{0}^{6} + bT_{1}^{6}, 6. Order 8

• D_{8}(a_{3}) g = [it_{0}, t_{1}, −it_{2}, ±ϵ_{8}t_{3}]

F = T_{3}^{2} + T_{2}^{3} + aT_{0}^{2}T_{1}^{2}T_{2} + T_{0}T_{1}(T_{0}^{4} + T_{1}^{4}),

7. Order 10 • E_{8}(a_{6}) g = [t_{0}, ϵ_{5}t_{1}, t_{2}, −t_{3}]

F = T_{3}^{2} + T_{2}^{3} + aT_{0}^{4}T_{2} + T_{0}(bT_{0}^{5} + T_{1}^{5}),

8. Order 12 • E_{8}(a_{3}) g = [−t_{0}, t_{1}, ϵ_{6}t_{2}, it_{3}]

F = T_{3}^{2} + T_{2}^{3} + T_{0}T_{1}(T_{0}^{4} + aT_{0}^{2}T_{1}^{2} + T_{1}^{4}),

9. Order 15 • E_{8}(a_{5}) g = [t_{0}, ϵ_{5}t_{1}, ϵ_{3}t_{2}, t_{3}]

F = T_{3}^{2} + T_{2}^{3} + T_{0}(T_{0}^{5} + T_{1}^{5}),

10. Order 20 • E_{8}(a_{2}) g = [t_{0}, ϵ_{10}t_{1}, −t_{2}, it_{3}]

F = T_{3}^{2} + T_{2}^{3} + aT_{0}^{4}T_{2} + T_{0}T_{1}^{5},

11. Order 24 • E_{8}(a_{1}) g = [it_{0}, t_{1}, ϵ_{12}t_{2}, ϵ_{8}t_{3}]

F = T_{3}^{2} + T_{2}^{3} + T_{0}T_{1}(T_{0}^{4} + T_{1}^{4}),

12. Order 30 • E_{8} g = [t_{0}, ϵ_{5}t_{1}, ϵ_{3}t_{2}, −t_{3}]

F = T_{3}^{2} + T_{2}^{3} + T_{0}(T_{0}^{5} + T_{1}^{5}).

To list all minimal subgroups of Aut(S) is very easy. We know that any subgroup in Ker(r) contains contains one of the elements α, β, αβ which are all minimal of types 8A_{1}, 4A_{2}, E_{8}(a_{8}). So, a subgroup is not minimal only if its image P in Aut(B) can be lifted isomorphically to Aut(S).

We will use the following lemma.

Lemma 6.18. Let P ⊂ Aut(ℙ^{1}) and G ⊂ Aut(S) be contained in r^{−1}(P). Then G is a minimal group unless G = P̃ ≅ P and G is a non-minimal cyclic group or non-minimal dihedral group D_{6}.

Proof. It follows from above classification of possible subgroups of Aut(B) and its lifts to Aut(S) that any non-isomorphic lift contains β or α, or βα which generate minimal cyclic groups. If the lift is isomorphic to P then P is either a cyclic group or P ≅ D_{6}. The group D_{6} contains 3 elements of type 4A_{1} and 2 elements of type 2A_{2}. Adding up the traces we see that the group is not minimal.

Let us classify minimal groups of automorphisms of a Del Pezzo surface of degree 1. As in the previous cases, to find a structure of such groups is enough to consider the types of surfaces which are not specialized to surfaces of other types. The following notation Type A→Type B indicates that a surface of type A specializes to a surface of type B.

V, IX, XIV, XV I, XV II, XIX, XXI → I,
III, V I, X, XI, XII, XV I, XV II, XV III, XX, XXI → II
XIII, XXI → IV, XIII, XXI → V III,
XII, XX, XXI → V II, XX, XI → XV.

It remains to consider surfaces of types

I, II, IV, V II, V III, XV.

Type I. P ≅ S_{4}.

Possible conjugacy classes of subgroups H are {1}, 2, 2, 3, 2^{2}, 4, D_{8}, D_{6}, A_{4}, S_{4}. Groups of order 2 have two conjugacy classes in P represented by [it_{0}, −it_{1}] and [−t_{1}, t_{0}]. The groups are realized in cases (iii) and (iv). None of them lifts isomorphically. An cyclic group of order 3 is generated by a non-minimal element realized in case (vii). Its isomorphic minimal lift is not minimal. A cyclic group of order 4 does not admit an isomorphic lift. The dihedral subgroup 2^{2} is of type (ii). This information, together with Lemma 6.18 allows us to classify all minimal subgroups.

• P = {1}: ⟨βα⟩ ≅ 6, ⟨α⟩ ≅ 3, ⟨α⟩ ≅ 2;

• P = 2: 4, 12;

• P = 2: 4, 12;

• P = 3: 3^{2}, 3 × 6;

• P = 2^{2}: Q_{8}, Q_{8} × 3;

• P = 2^{2}: D_{8}, D_{8} × 3;

• P = 4: 8, 8 × 3;

• P = D_{8}: D_{16}, D_{8} × 3;

• P = D_{6}: D_{6} × 2, D_{6} × 3, D_{6} × 6;

• P = A_{4}: \overline{T}, \overline{T} × 3;

• P = S_{4}: \overline{T} : 2, 3 × (\overline{T} : 2).

Surfaces specializing to a surface of type I have the following minimal subgroups.

V: 4, 6, Q_{8}, \overline{T} .

IX: 4(2), 8, D_{16}.

XIV: 4, Q_{8}.

XVI: D_{8}.

XVII:2, 3, 6.

XIX: 2, 4.

XXI: 2.

Type II: P = D_{12}.

Possible subgroups are {1}, 2, 2, 3, 2^{2}, 6, D_{6}, D_{12}. Cyclic subgroups of order 2, 3 and 6 admit isomorphic non-minimal lifts. All these groups are not minimal. There are two conjugacy classes of subgroups of order 2 in P represented by [it_{0}, −it_{1}] and [−t_{1}, t_{0}]. One subgroups lifts isomorphically, other one does not. The cyclic group of order 6 admits an isomorphic lift and not minimal. The dihedral group D_{6} admits a non-minimal isomorphic lift.

• P = {1}: ⟨βα⟩ ≅ 6, ⟨α⟩ ≅ 3, ⟨α⟩ ≅ 2;

• P = 2: 4, 12;

• P = 2: 2^{2}, 2^{2} × 3, 6;

• P = 3: 3^{2}, 3^{2} × 2;

• P = 2^{2}: Q_{8}, Q_{8} × 3;

• P = 6: 2 × 6,

• P = D_{6}: 2 × D_{6}, D_{6} × 3, D_{6} × 6;

• P = D_{12}: 2D_{12}, 3 × 2D_{12}.

Surfaces specializing to a surface of type II have the following subgroups: III: 4, 12, 2^{2}, 2^{2} × 3, 6, 3^{2}, 3^{2} × 2, Q_{8}, Q_{8} × 3, 2 × D_{6}, D_{6} × 3, D_{6} × 6. VI: 4, 2^{2}, 3^{2}, Q_{8}, 2 × 6, 2 × D_{6}, 2D_{12} ≅ (2 × 6)^{•}2. X: 2, 2 × D_{6}. XI: 2, 3, 6, 2^{2}, 2 × 6. XII: 6 × 2, 6, 2^{2}, 2 × 6. XVI: 2, 4, D_{8}. XVII: 2, 3, 6. XVIII: 2, 6. XX: 2, 2^{2}. XXI: 2. Type IV: P = 5 This is easy. We have P ≅ 5. It admits an isomorphic lift

to a non-minimal subgroup.

• P = {1}: ⟨βα⟩ ≅ 6, ⟨α⟩ ≅ 3, ⟨α⟩ ≅ 2;

• P = 5: 5, 10, 15, 30;

Surfaces specializing to a surface of type IV have the following subgroups: XIII: 5, 10. XXI: 2. Type VII: P ≅ 12.

• P = 2: 2^{2}.

• P = 3: 6;

• P = 4: 2 × 4;

• P = 6: 2 × 6;

• P = 12: 2 × 12.

Surfaces specializing to a surface of type VII have the following subgroups: XII: 2, 6, 2 × 6. XX: 2, 2^{2}. XXI: 2. Type VIII: P ≅ 10.

• P = {1}: ⟨βα⟩ ≅ 6, ⟨α⟩ ≅ 3, ⟨α⟩ ≅ 2;

• P = 2: 2^{2}.

• P = 5: 10;

• P = 10: 20.

Surfaces specializing to a surface of type VIII have the following subgroups: XIII: 5, 10. XXI: 2. Type XV: P ≅ 4.

• P = {1}: ⟨βα⟩ ≅ 6, ⟨α⟩ ≅ 3, ⟨α⟩ ≅ 2;

• P = 2: 2^{2}.

• P = 4: 2 × 4.

Surfaces specializing to a surface of type VIII have the following subgroups: XX: 2, 2^{2}. XXI: 2.

7 Elementary links and factorization theorem

7.1 Noether-Fano inequality

Let |dℓ − m_{1}x_{1} − ⋯ − m_{N}x_{N}| be a homaloidal net in ℙ^{2}. The following is a well-known classical result.

Lemma 7.1. (Noether’s inequality) Assume d > 1, m_{1} ≥ ⋯ ≥ m_{N} ≥ 0. Then

m_{1} + m_{2} + m_{3} ≥ d + 1,

and the equality holds if and only if either m_{1} = ⋯ = m_{N} or m_{1} = n−1, m_{2} = ⋯ = m_{N}.

Proof. We have

m_{1}^{2} + ⋯ + m_{N}^{2} = d^{2} − 1, m_{1} + ⋯ + m_{N} = 3d − 3.

Multiplying the second equality by m_{3} and subtracting from the first one, we get

m_{1}(m_{1} − m_{3}) + m_{2}(m_{2} − m_{3}) − ∑_{i≥4} m_{i}(m_{3} − m_{i}) = d^{2} − 1 − 3m_{3}(d − 1).

¿From this we obtain

(d−1)(m_{1}+m_{2}+m_{3}−d−1) = (m_{1}−m_{3})(d−1−m_{1})+(m_{2}−m_{3})(d−1−m_{2})+

+ ∑_{i≥4} m_{i}(m_{3} − m_{i}).

Since d − 1 − m_{i} ≥ 0, this obviously proves the assertion.

Corollary 7.2.

m_{1} > d/3.

Let us generalize Corollary 7.2 to birational maps of any rational surfaces. The same idea works even for higher-dimensional varieties. Let χ : S → S′ be a birational map of surfaces. Let σ : X → S, ϕ : X → S′ be its resolution. Let |H′| be a linear system on S′ without base points. Let

ϕ^{∗}(H′) ∼ σ^{∗}(H) − ∑_{i} m_{i}E_{i}

for some divisor H on S and exceptional curves E_{i} of the map σ.

Theorem 7.3. (Noether-Fano inequality) Assume that there exists some integer m_{0} ≥ 0 such that |H′ + mK_{S′}| = ∅ for m ≥ m_{0}. For any m ≥ m_{0} such that |H + mK_{S}| ≠ ∅ there exists i such that

m_{i} > m.

Proof. We know that K_{X} = σ^{∗}(K_{S}) + ∑_{i} E_{i}. Thus we have the equality in Pic(X)

ϕ^{∗}(H′) + mK_{X} = (σ^{∗}(H + mK_{S})) + ∑(m − m_{i})E_{i}.

Applying f_{∗} to the left-hand side we get the divisor class H′ + mK_{S′} which, by assumption, cannot be effective. Since |σ^{∗}(H + mK_{S})| ≠ ∅, applying ϕ_{∗} to the right-hand side, we get the sum of an effective divisor and the image of the divisor ∑_{i}(m − m_{i})E_{i}. If all m − m_{i} are nonnegative, it is also an effective divisor, and we get a contradiction. Thus there exists i such that m − m_{i} < 0.

Example 7.4. Assume S = S′ = ℙ^{2}, H = dℓ and H′ = ℓ. We have |H + K_{S′}| = | − 2ℓ| = ∅. Thus we can take m_{0} = 1. If d ≥ 3, we have for any 1 ≤ a ≤ d/3, |H′ + aK_{S}| = |(d − 3a)ℓ| ≠ ∅. This gives m_{i} > d/3 for some i. This is Corollary 7.2.

Example 7.5. Let χ : S− → S′ be a birational map of Del Pezzo surfaces. Assume that S′ is not a quadric or the plane. Consider the complete linear system H′ = |−K_{S′}|. Then |H′+mK_{S′}| = ∅ for m ≥ 2. Let χ^{−1}(H′) = |D−η| be its proper transform on S. Choose a standard basis (e_{0}, …, e_{k}) in Pic(S) corresponding to the blow-up S → ℙ^{2}. Since K_{S} = −3e_{0} + e_{1} + ⋯ + e_{k}, we can write χ^{−1}(H′) = | − aK_{S} − ∑ m_{i}x_{i}|, where a ∈ \frac{1}{3}ℤ. Assume that χ^{∗}(H′) = −aK_{S}. Then there exists a point with multiplicity ≥ a if a > 1 that we assume.

Remark 7.6. The Noether inequality is of course well-known (see, for example, [2], [34]). We give it here to express our respect of this important and beautiful result of classical algebraic geometry. Its generalization from Theorem 7.3 is also well-known (see [38], 1.3). Note that the result can be also applied to G-equivariant maps χ provided that the linear system |H′| is Ginvariant. In this case the linear system |H − η| is also G-invariant and the bubble cycle η = ∑ m_{i}x_{i} consists of the sum of G-orbits.

The existence of base points of high multiplicity in the linear system |H − η| = χ^{−1}(H′) follows from the classical theory of termination of the adjoint system for rational surfaces which goes back to G. Castelnuovo. Nowadays this theory has an elegant interpretation in the Mori theory which we give in the next section.

7.2 Elementary links

We will be dealing with minimal Del Pezzo G-surfaces or minimal conic bundles G-surfaces. In the G-equivariant version of the Mori theory they are interpreted as extremal contractions ϕ : S → C, where C = pt is a point in the first case and C ≅ ℙ^{1} in the second case. They are also two-dimensional analogs of rational Mori G-fibrations.

A birational G-map between the Mori fibrations are diagrams

\xymatrix{S \ar[d]_{ϕ} \ar@{-->}[r]^{χ} & S′ \ar[d]_{ϕ′} \\ C & C′}  (51)

which in general do not commute with the fibrations. These maps are decomposed into elementary links. These links are divided into the four following types.

• Links of type I:

They are commutative diagrams of the form

\xymatrix{S \ar[d]_{ϕ} & Z = S′ \ar[d]_{ϕ′} \ar[l]_{σ} \\ C = pt & C′ = ℙ^{1} \ar[l]_{α}}  (52)

Here σ : Z → S is the blow-up of a G-orbit, S is a minimal Del Pezzo surface, ϕ′ : S′ → ℙ^{1} is a minimal conic bundle G-fibration, α is the constant map. For example, the blow-up of a G-fixed point on ℙ^{2} defines a minimal conic G-bundle ϕ′ : F_{1} → ℙ^{1} with a G-invariant exceptional section.

• Links of type II:

They are commutative diagrams of the form

\xymatrix{S \ar[d]_{ϕ} & Z \ar[l]_{σ} \ar[r]^{τ} & S′ \ar[d]_{ϕ′} \\ C & = & C′}  (53)

Here σ : Z → S, τ : Z → S′ are the blow-ups of G-orbits such that rank Pic(Z)^{G} = rank Pic(S)^{G} + 1 = rank Pic(S′)^{G} + 1, C = C′ is either a point or ℙ^{1}. An example of a link of type II is the Geiser (or Bertini) involution of ℙ^{2}, where one blows up 7 (or 8) points in general position which form one G-orbit. Another frequently used link of type II is an elementary transformation of minimal ruled surfaces and conic bundles.

• Links of type III:

These are the birational maps which are the inverses of links of type I.

• Links of type IV:

They exist when S has two different structures of G-equivariant conic bundles. The link is the exchange of the two conic bundle structures

\xymatrix{S \ar[d]_{ϕ} & = & S′ \ar[d]_{ϕ′} \\ C &  & C′}  (54)

One uses these links to relate elementary links with respect to one conic fibration to elementary links with respect to another conic fibration. Often the change of the conic bundle structures is realized via an involution in Aut(S), for example, the switch of the factors of S = ℙ^{1} × ℙ^{1} (see the following classification of elementary links).

7.3 The factorization theorem

Let χ : S− → S′ be a G-equivariant birational map of minimal G-surfaces. We would like to decompose it into a composition of elementary links. This is achieved with help of G-equivariant theory of log-pairs (S, D), where D is a G-invariant ℚ-divisor on S. It is chosen as follows. Let us fix a G-invariant very ample linear system H′ on S′. If S′ is a minimal Del Pezzo surface we take H′ = |−a′K_{S′}|, a′ ∈ ℤ_{+}. If S′ is a conic bundle we take H′ = |−a′K_{S′} +b′f′|, where f′ is the class of a fibre of the conic bundle, a′, b′ are some appropriate positive integers.

Let H = H_{S} = χ^{−1}(H′) be the proper transform of H′ on S. Then

H = | − aK_{S} − ∑ m_{x}x|,

if S is a Del Pezzo surface, a ∈ \frac{1}{2}ℤ_{+} ∪ \frac{1}{3}ℤ_{+}, and

H = | − aK_{S} + bf − ∑ m_{x}x|,

if S is a conic bundle, a ∈ \frac{1}{2}ℤ_{+}, b ∈ \frac{1}{2}ℤ. The linear system H is G-invariant, and the 0-cycle ∑ m_{x}x is a sum of G-orbits with integer multiplicities. One uses the theory of log-pairs (S, D), where D is a general divisor from the linear system H, by applying some “untwisting links” to χ in order to decrease the number a, the algebraic degree of H. Since a is a rational positive number with bounded denominator, this process terminates after finitely many steps (see [21],[38]).

Theorem 7.7. Let f : S− → S′ be a birational map of minimal G-surfaces. Then χ is equal to a composition of elementary links.

The proof of this theorem is the same as in the arithmetic case ([38], Theorem 2.5). Each time one chooses a link to apply and the criterion used for termination of the process is based on the following version of Noether’s inequality in the Mori theory.

Lemma 7.8. In the notation from above, if m_{x} ≤ a for all base points x of H and b ≥ 0 in the case of conic bundles, then χ is an isomorphism.

The proof of this lemma is the same as in the arithmetic case ([38], Lemma 2.4).

We will call a base points x of H with m_{x} > a a maximal singularity of H. It follows from 3.2 that if H has a maximal singularity of height > 0, then it also has a maximal singularity of height 0. We will be applying the “untwisting links” of types I-III to these points. If ϕ : S → ℙ^{1} is a conic bundle with all its maximal singularities untwisted with helps of links of type II, then either the algorithm terminates, or b < 0. In the latter case the linear system |K_{S} + \frac{1}{a}H| = |\frac{b}{a}f| is not nef and has canonical singularities (i.e. no maximal singularities). Applying the theory of log-pairs to the pair (S, |\frac{b}{a}f|) we find an extremal contraction ϕ′ : S → ℙ^{1}, i.e. another conic bundle structure on S. Rewriting H in a new basis −K_{S}, f′ we find the new coefficient a′ < a. Applying the link of type IV relating ϕ and ϕ′, we start over the algorithm with decreased a.

It follows from the proofs of Theorem 7.7 and Lemma 7.8 that all maximal singularities of H are in general position in the following sense.

(i) If S is a minimal Del Pezzo G-surface, then the blow-up of all maximal singularities of H is a Del Pezzo surface (of course this agrees with the description of points in general position at the end of section 3.8).

(ii) If ϕ : S → ℙ^{1} is a conic bundle, then none of the maximal singularities lie on a singular fibre of ϕ and no two lie on one fibre.

The meaning of these assertions is that the linear system |H| has no fixed components. In the case of Del Pezzo surfaces with an orbit of maximal singular points we can find a link by blowing up this orbit to obtain a surface Z with Pic(Z)^{G} ≅ ℤ⊕ℤ and two extremal rays. By applying Kleiman’s criterion this implies that −K_{Z} is ample. The similar situation occurs in the case of conic bundles (see [38], Comment 2).

Let S be a minimal Del Pezzo G-surface of degree d. Let us write H_{S} = | − aK_{S} − ∑ m_{κ}κ| as in (16).

Lemma 7.9. Let κ_{1}, …, κ_{n} be the G-orbits of maximal multiplicity. Then

∑_{i=1}^{n} d(κ_{i}) < d.

Proof. Let D_{1}, D_{2} ∈ H_{S} be two general divisors from H_{S}. Since H_{S} has no fixed components, we have

0 ≤ D_{1} · D_{2} = a^{2}d − ∑ m_{κ}^{2}d(κ) ≤ a^{2}d − ∑_{i=1}^{n} m_{κ_{i}}^{2}d(κ_{i}) =

a^{2}(d − ∑_{i=1}^{n} d(κ_{i})) − ∑_{i=1}^{n}(m_{κ_{i}}^{2} − a^{2})d(κ_{i}).

It follows from Example 7.5 that m_{κ_{i}} > a for all i = 1, …, n. This implies that d > ∑_{i=1}^{n} d(κ_{i}).

Definition 7.10. A minimal Del Pezzo G-surface is called superrigid (resp. rigid) if any birational G-map χ : S− → S′ is a G-isomorphism (resp. there exists a birational G-automorphism α : S− → S such that χ ∘ α is a Gisomorphism).

A minimal conic bundle ϕ : S → ℙ^{1} is called superrigid (resp. rigid) if for any birational G-map χ : S− → S′, where ϕ′ : S′ → ℙ^{1} is a minimal conic bundle, there exists an isomorphism δ : ℙ^{1} → ℙ^{1} such that the following diagram is commutative

\xymatrix{S \ar[d]_{ϕ} \ar@{-->}[r]^{χ} & S′ \ar[d]_{ϕ′} \\ ℙ^{1} \ar[r]^{δ} & ℙ^{1}}  (55)

(resp. there exists a birational G-automorphism α : S− → S′ such that the diagram is commutative after we replace χ with χ ∘ α).

Applying Lemma 7.8 and Lemma 7.9, we get the following.

Corollary 7.11. Let S be a minimal Del Pezzo G-surface of degree d = K_{S}^{2}. If S has no G-orbits κ with d(κ) < d, then S is superrigid. In particular, a Del Pezzo surface of degree 1 is always superrigid and a Del Pezzo surface of degree 2 is superrigid unless G has a fixed point.

A minimal conic G-bundle with K_{S}^{2} ≤ 0 is superrigid.

The first assertion is clear. To prove the second one, we untwist all maximal base points of H_{S} with help of links of type II to get a conic bundle ϕ′ : S′ → ℙ^{1} with b′ < 0. Since H_{S′}^{2} = a^{2}K_{S′}^{2} + 4ab′ − ∑ m′_{x}^{2} ≥ 0 and K_{S′}^{2} = K_{S}^{2} ≤ 0, 4ab′ < 0, we get a contradiction with Lemma 7.8. Thus χ after untwisting maximal base points terminates at an isomorphism (see [35], [36], [38], Theorem 1.6).

7.4 Classification of elementary links

Here we consider an elementary link f : S− → S′ defined by a resolution (S ←^{σ} Z →^{τ} S′). We take H_{S′} to be the linear system | − aK_{S′}| if S′ is a Del Pezzo surface and |f| if S′ is a conic bundle, where f is the divisor class of a fibre. It is assumed that the point which we blow up are in general position in sense of the previous subsection.

We denote by DP_{k} (resp. C_{k}) the set of isomorphism classes of minimal Del Pezzo surfaces (resp. conic bundles) with k = K_{S}^{2} (resp. k = 8 − K_{S}^{2}).

Proposition 7.12. Let S, S′ be as in Link I of type I. The map σ : Z = S′ → S is the blowing up of a G-invariant bubble cycle η with ht(η) = 0 of some degree d. The proper transform of the linear system |f| on S′ is equal to the linear system H_{S} = | − aK_{S} − mη|. Here f is the class of a fibre of the conic bundle structure on S′. The following cases are possible:

1. K_{S}^{2} = 9 • S = ℙ^{2}, S′ = F_{1}, d = 1, m = 1, a = \frac{1}{3}.

• S = ℙ^{2}, S′ ∈ C_{3}, d = 4, m = 1, a = \frac{2}{3}.

2. K_{S}^{2} = 8 • S = F_{0}, π : S′ → ℙ^{1} a conic bundle with k = 2, d = 2, m = 1, a = \frac{1}{2}.

3. K_{S}^{2} = 4

• S ∈ DP_{4}, p : S′ → ℙ^{1} a conic bundle with f = −K_{S′} − l, where l is a (-1)-curve, d = a = 1, m = 2.

Proof. Let H_{S} = |−aK_{S} −bη|, where η is a G-invariant bubble cycle of degree d. We have

(−aK_{S} − bη)^{2} = a^{2}K_{S}^{2} − b^{2}d = 0, (−aK_{S} − bη, −K_{S}) = aK_{S}^{2} − bd = 2.

Let t = b/a. We have

(td)^{2} = dK_{S}^{2}, K_{S}^{2} − td = 2/a > 0.

The second inequality, gives td < K_{S}^{2}, hence d < K_{S}^{2}. Giving the possible values for K_{S}^{2} and using that a ∈ \frac{1}{3}ℤ, we check that the only possibilities are:

(K_{S}^{2}, d, t) = (9, 1, 3), (8, 2, 2), (4, 1, 2), (4, 2, 1).

This gives our cases and one extra case (4, 2, 1). In this case a = 2 and H_{S} = | − 2K_{S} − 2x_{1}| contradicting the primitiveness of f. Note that this case is realized in the case when the ground field is not algebraically closed (see [38]).

Proposition 7.13. Let S, S′ be as in Link of type II. Assume that S, S′ are both minimal Del Pezzo surfaces. Then (S ←^{σ} Z →^{τ} S′), where σ is the blow-up of a G-invariant bubble cycle η with ht(η) = 0 and some degree d. The proper transform of the linear system | − K_{S′}| on S is equal to | − aK_{S} − mη|. And similarly defined d′, m′, a′ for τ. The following cases are possible:

1. K_{S}^{2} = 9

• S′ ≅ S = ℙ^{2}, d = d′ = 8, m = m′ = 18, a = a′ = 17 (S ← Z → S′) is a minimal resolution of a Bertini involution).

• S′ ≅ S = ℙ^{2}, d = d′ = 7, m = m′ = 9, a = a′ = 8 (S ← Z → S′) is a minimal resolution of a Geiser transformation).

• S′ ≅ S = ℙ^{2}, d = d′ = 6, m = m′ = 6, a = a′ = 5 (S ← Z → S′) is a minimal resolution of a Cremona transformation given by the linear system |5ℓ − 2p_{1} − 2p_{2} − 2p_{3} − 2p_{4} − 2p_{5}|,

• S ≅ ℙ^{2}, S′ ∈ DP_{5}, d = 5, m′ = 6, a = \frac{5}{3}, d′ = 1, m = 2, a′ = 3.

• S ≅ S′ = ℙ^{2}, d = d′ = 3, m = m′ = 1, a = a′ = \frac{2}{3}, (S ← Z → S′) is a minimal resolution of a standard quadratic transformation.

• S = ℙ^{2}, S′ = F_{0}, d = 2, m = 3, a′ = \frac{3}{2}, d′ = 1, a = \frac{4}{3}.

2. K_{S}^{2} = 8 • S ≅ S′ ≅ F_{0}, d = d′ = 7, a = a′ = 15, m = m′ = 16.

• S ≅ S′ ≅ F_{0}, d = d′ = 6, a = a′ = 7, m = m′ = 8.

• S ≅ F_{0}, S′ ∈ DP_{5}, d = 5, d′ = 2, a = \frac{5}{2}, m = 4, a′ = 4, m′ = 6.

• S ≅ F_{0}, S′ ≅ F_{0}, d = d′ = 4, a = a′ = 3, m = m′ = 4.

• S ≅ F_{0}, S′ ∈ DP_{6}, d = 3, d′ = 1, a = \frac{3}{2}, m = 2, m′ = 4, a′ = 2.

• S ≅ F_{0}, S′ ≅ ℙ^{2}, d = 1, d′ = 2, a = \frac{3}{2}, m = 3, a′ = \frac{4}{3}, m′ = 2. This link

is the inverse of the last case from the preceding list.

3. K_{S}^{2} = 6 • S ≅ S′ ∈ DP_{6}, d = d′ = 5, a = 11, m = 12.

• S ≅ S′ ∈ DP_{6}, d = d′ = 4, a = 5, m = 6.

• S ≅ S′ ∈ DP_{6}, d = d′ = 3, a = 3, m = 4.

• S ≅ S′ ∈ DP_{6}, d = d′ = 2, a = 2, m = 3.

• S ∈ DP_{6}, S′ = F_{0}, d = 1, d′ = 3, a = \frac{3}{2}, m = 2. This link is the inverse

of the link from the preceding list with S′ ∈ DP_{6}, d = 3.

4. K_{S}^{2} = 5 • S ≅ S′ ∈ DP_{5}, d = d′ = 4, m = m′ = 10, a = a′ = 5..

• S = S′ ∈ DP_{5}, d = d′ = 3, m = m′ = 5, a = a′ = 4.

• S ∈ DP_{5}, S′ = F_{0}, d = 2, d′ = 5. This link is inverse of the link with S = F_{0}, S′ ∈ DP_{5}, d = 5.

• S ∈ DP_{5}, S′ = ℙ^{2}, d = 1, d′ = 5. This link is inverse of the link with

S = ℙ^{2}, S′ ∈ DP_{5}, d = 5.

5. K_{S}^{2} = 4 • S ≅ S′ ∈ DP_{4}, d = d′ = 3. This is an analog of the Bertini involution.

• S ≅ S′ ∈ DP_{4}, d = d′ = 2. This is an analog of the Geiser involution.

6. K_{S}^{2} = 3 • S ≅ S′ ∈ DP_{3}, d = d′ = 2. This is an analog of the Bertini involution.

• S ≅ S′ ∈ DP_{3}, d = d′ = 1. This is an analog of the Geiser involution.

7. K_{S}^{2} = 2 • S = S′ ∈ DP_{2}, d = d′ = 1. This is an analog of the Bertini involution.

Proof. Similar to the proof of the previous proposition, we use that

H_{S}^{2} = a^{2}K_{S}^{2} − b^{2}d = K_{S′}^{2}, aK_{S}^{2} − bd = K_{S′}^{2},
H_{S′}^{2} = a′^{2}K_{S′}^{2} − b′^{2}d = K_{S}^{2}, a′K_{S}^{2} − b′d = K_{S}^{2}.

Since the link is not a biregular map, by Noether’s inequality we have a > 1, a′ > 1, b > a, b′ > a′. This implies

d < K_{S}^{2} − \frac{1}{a}K_{S′}^{2}, d′ < K_{S′}^{2} − \frac{1}{a′} K_{S}^{2}.

It is not difficult to list all solutions. For example, assume K_{S}^{2} = 1. Since d is a positive integer, we see that there are no solutions. If K_{S}^{2} = 2, we must have d = d′ = 1.

Proposition 7.14. Let S, S′ be as in Link of type II. Assume that S, S′ are both minimal conic bundles. Then (S ← Z → S′) is a composition of elementary transformations elm_{x_{1}} ∘ … ∘ elm_{x_{s}}, where (x_{1}, …, x_{s}) is a G-orbit of points not lying on a singular fibre with no two points lying on the same fibre.

We skip the classification of links of type III. They are the inverses of links of type I.

Proposition 7.15. Let S, S′ be as in Link of type IV. Recall that they consist of changing the conic bundle structure. The following cases are possible:

• K_{S}^{2} = 8, S′ = S, f′ = −K_{S′}−f, it is represented by a switch automorphism;

• K_{S}^{2} = 4, S′ = S, f′ = −K_{S′} − f;

• K_{S}^{2} = 2, S′ = S, f′ = −2K_{S′} − f; it is represented by a Geiser involution;

• K_{S}^{2} = 1, S′ = S, f′ = −4K_{S′} − f; it is represented by a Bertini involution;

Proof. In this case S admits two extremal rays and rank Pic(S)^{G} = 2 so that −K_{S} is ample. Let |f′| be the second conic bundle. Write f′ ∼ −aK_{S} + bf. Using that f′^{2} = 0, f · K_{S} = f′ · K_{S} = −2, we easily get b = −1 and aK_{S}^{2} = 4. This gives all possible cases from the assertion.

8 Birational classes of minimal G-surfaces

8.1

Let S be a minimal G-surface S and d = K_{S}^{2}. We will classify all isomorphism classes of (S, G) according to the increasing parameter d. Since the number of singular fibres of a minimal conic bundle is at least 4, we have d ≤ 4 for conic bundles.

• d ≤ 0.

By Corollary 7.11, S is a superrigid conic bundle with k = 8 − d singular fibres. The number k is a birational invariant. The group G is of de Jonquiéres type and its conjugacy class in Cr(2) is determined uniquely by Theorem 5.7 or Theorem 5.3.

Also observe that if ϕ : S → ℙ^{1} is an exceptional conic bundle and G_{0} = Ker(G → O(Pic(S)) is non-trivial, then no links of type II is possible. Thus the conjugacy class of G is uniquely determined by the isomorphism class of S.

• d = 1, S is a Del Pezzo surface.

By Corollary 7.11, the surface S is superrigid. Hence the conjugacy class of G is determined uniquely by its conjugacy class in Aut(S). All such conjugacy classes are listed in subsection 6.7.

• d = 1, S is a conic bundle.

Let ϕ : S → ℙ^{1} be a minimal conic bundle on S. It has k = 7 singular fibres. If −K_{S} is ample, i.e. S is a (non-minimal) Del Pezzo surface, then the center of Aut(S) contains the Bertini involution β. We know that β acts as −1 on R_{S}, thus β does not act identically on Pic(S)^{G}, hence β ∉ G. Since k is odd, the conic bundle is not exceptional, so we cam apply Theorem 5.7, Case (2). It follows that G must contain a subgroup isomorphic to 2^{2}, adding β we get that S is a minimal Del Pezzo 2^{3}-surface of degree 1. However, the classification shows that there are no such surfaces.

Thus −K_{S} is not ample. It follows from Proposition 7.13 that the structure of a conic bundle on S is unique. Any other conic bundle birationally Gisomorphic to S is obtained from S by elementary transformations with Ginvariant set of centers.

• d = 2, S is a Del Pezzo surface.

By Corollary 7.11, S is superrigid unless G has a fixed point on S. If χ : S− → S′ is a birational G-map, then H_{S} has only one maximal base point and it does not lie on a (−1)-curve. We can apply an elementary link Z → S, Z → S of type II which together with the projections S → ℙ^{2} resolves the Bertini involution. These links together with the G-automorphisms (including the Geiser involution) generate the group of birational G-automorphisms of S (see [38], Theorem 4.6). Thus the surface is rigid. The conjugacy class of G in Cr(2) is determined uniquely by the conjugacy class of G in Aut(S). All such conjugacy classes are listed in Table 7 and Theorem 6.17.

• d = 2, ϕ : S → ℙ^{1} is a conic bundle.

If −K_{S} is ample, then ϕ is not exceptional. The center of Aut(S) contains the Geiser involution γ. Since γ acts non-trivially on Pic(S)^{G} = ℤ^{2}, we see that γ ∉ G. Applying γ we obtain another conic bundle structure. In other words, γ defines an elementary link of type IV. Using the factorization theorem we show that the group of birational G-automorphisms of S is generated by links of type II, the Geiser involution, and G-automorphisms (see [36], [39], Theorem 4.9). Thus ϕ : S → ℙ^{1} is a rigid conic bundle.

If S is not a Del Pezzo surface, ϕ could be an exceptional conic bundle with g = 2. In any case the group G is determined in Theorem 5.3. We do not know whether S can be mapped to a conic bundle with −K_{S} ample (see [36]).

Applying Proposition 5.2, we obtain that any conic bundle with d ≥ 3 is a non-minimal Del Pezzo surface, unless d = 4 and S is an exceptional conic bundle. In the latter case, the group G can be found in Theorem 5.3. It is not known whether it is birationally G-isomorphic to a Del Pezzo surface. It is true in the arithmetic case.

• d = 3, S is a minimal Del Pezzo surface.

The classification of elementary links shows that S is rigid. Birational Gautomorphisms are generated by links of type (6) from Proposition 7.12. The conjugacy class of G in Cr(2) is determined by the conjugacy class of G in Aut(S).

• d = 3, S is a minimal conic bundle.

Since k = 5 is odd, G has 3 commuting involutions, the fixed-point locus of one of them must be a rational 2-section of the conic bundle. It is easy to see that it is a (−1)-curve C from the divisor class −K_{S} − f. The other two fixed-point curves are of genus 2. The group G leaves the curve C invariant. Thus blowing it down, we obtain a minimal Del Pezzo G-surface S′ of degree 4. The group G contains a subgroup isomorphic to 2^{2}. Thus G can be found in the list of minimal groups of degree 4 Del Pezzo surfaces with a fixed point. For example, the group 2^{2} has 4 fixed points.

• d = 4, S is a minimal Del Pezzo surface.

If S^{G} = ∅, then S admits only self-links of type II, so it is rigid or superrigid. The conjugacy class of G in Cr(2) is determined by the conjugacy class of G in Aut(S) and we can apply Theorem 6.9. If x is a fixed point of G, then we can apply a link of type I, to get a minimal conic bundle with d = 3. So, all groups with S^{G} ≠ ∅ are conjugate to groups of de Jonquiéres type realized on a conic bundle S ∈ C_{5}.

• d = 4, S is a minimal conic bundle.

Since k = 4, it follows from Lemma 5.1 that either S is an exceptional conic bundle with g = 1, or S is a Del Pezzo surface with two sections with selfintersection −1 intersecting at one point. In the latter case, S is obtained by regularizing a de Jonquéres involution IH_{3} (see section 2.3). In the case when S is an exceptional conic bundle the groups of automorphisms are described in Example 5.4. They are minimal if and only if the kernel of the map G → PGL(2) contains an involution not contained in G_{0} = Ker(G → O(Pic(S)). If G_{0} is not trivial, then no elementary transformation is possible. So, S is not birationally isomorphic to a Del Pezzo surface.

• d = 5, S is a Del Pezzo surface, G ≅ 5.

Let us show that (S, G) is birationally isomorphic to (ℙ^{2}, G). Since rational surfaces are simply-connected, G has a fixed point x on S. The anti-canonical model of S is a surface of degree 5 in ℙ^{5}. Let P be the tangent plane of S at x. The projection from P defines a birational G-equivariant map S− → ℙ^{2} given by the linear system of anti-canonical curves with double point at x. It is an elementary link of type II.

• d = 5, S is a Del Pezzo surface, G ≅ 5 : 2, 5 : 4.

The cyclic subgroup of order 5 of G has two fixed points on S. This immediately follows from the Lefschetz fixed-point formula. Since it is normal in G, the groups G has an orbit κ with d(κ) = 2. Using an elementary link of type II with S′ = F_{0}, we obtain that G is conjugate to a group acting on F_{0}.

• d = 5, S is a Del Pezzo surface, G ≅ A_{5}, S_{5}.

It is clear that S^{G} = ∅ since otherwise G admits a faithful 2-dimensional linear representation. It is known that it does not exist. Since A_{5} has no index 2 subgroups G does not admit orbits κ with d(κ) = 2. The same is obviously true for G = S_{5}. It follows from the classification of links that (S, G) is superrigid.

• d = 6.

One of the groups from this case, namely G ≅ 2 × S_{3} was considered in [39],

[40] (the papers also discuss the relation of this problem to some questions in the theory of algebraic groups raised in [43]). It is proved there that (S, G) is not birationally isomorphic to (ℙ^{2}, G) but birationally isomorphic to minimal (F_{0}, G). The birational isomorphism is easy to describe. We know that G contains the lift of the standard Cremona involution. It has 4 fixed points in S, the lifts of the points given in affine coordinates by (±1, ±1). The group S_{3} fixes (1, 1) and permutes the remaining points p_{1}, p_{2}, p_{3}. The proper transforms of the lines ⟨p_{i}, p_{j}⟩ in S are disjoint (−1)-curves E_{i}. The anti-canonical model of S is a surface of degree 6 in ℙ^{6}. The projection from the tangent plane to S at the fixed point, is a link of type II with S′ = F_{0}. It blows-up the fixed point, and then blows down the pre-images of the curves E_{i}. Now the group G acts on F_{0} with F_{0}^{G} = ∅.

If minimal G contains some non-trivial imprimitive projective transformations, then G has no fixed points. It follows from the classification of links that S is rigid. If G ≅ 6 or S_{3}, then it acts on F_{0} with a fixed point. The projection from this point defines a birational isomorphism (S, G) and (ℙ^{2}, G). Thus the only groups which are not conjugate to a group of projective transformations are the groups which are mapped surjectively onto W_{S} = S_{3} × 2. Those of them which are mapped isomorphically are conjugate to subgroups of F_{0}.

• d = 8.

In this case S = F_{0} or F_{n}, n > 1. In the first case (S, G) is birationally isomorphic to (ℙ^{2}, G) if S^{G} ≠ ∅ (via the projection from the fixed point). This implies that the subgroup G′ of G belonging to the connected component of the identity of Aut(F_{0}) is an extension of cyclic groups. As we saw in Theorem 4.9 this implies that G′ is an abelian group of transformations (x, y) ↦ (ϵ_{nk}^{a}x, ϵ_{mk}^{b}y), where a = sb mod k for some s coprime to k. If G ≠ G′, then we must have m = n = 1 and s = ±1 mod k.

If F_{0}^{G} = ∅ and Pic(F_{0})^{G} ≅ ℤ, then the classification of links shows that links of type II with d = d′ = 7, 6, 5, d = 3, d′ = 1 map F_{0} to 𝔽_{0} or to minimal Del Pezzo surfaces of degrees 5 or 6. These cases have been already considered. If rank Pic(S)^{G} = 2, then any birational G-map S− → S′ is composed of elementary transformations with respect to one of the conic bundle fibrations. They do not change K_{S}^{2} and do not give rise a fixed points. So, G is not conjugate to any subgroup of Aut(ℙ^{2}).

Assume n > 1. Then G = A.B, where A ≅ n acts identically on the base of the fibration and B ⊂ PGL(2). The subgroup B fixes pointwisely two disjoint sections, one of them is the exceptional one. Let us consider different cases corresponding to possible groups B.

B ≅ C_{n}. In this case B has two fixed points on the base, hence G has 2 fixed points on the non-exceptional section. Performing an elementary transformation with center at one of these points we descend G to a subgroup of F_{n−1}. Proceeding in this way, we arrive to the case n = 1, and then obtain that G is a group of automorphisms of ℙ^{2}.

B ≅ D_{n}. In this case B has an orbit of cardinality 2 in ℙ^{1}. A similar argument shows that G has an orbit of cardinality 2 on the non-exceptional section. Applying the product of the elementary transformations at these points we descend G to a subgroup of automorphisms of F_{n−2}. Proceeding in this way we obtain that G is a conjugate to a subgroup of Aut(ℙ^{2}) or of Aut(F_{0}). In the latter case it does not have fixed points, and hence is not conjugate to a linear subgroup of Cr(2).

B ≅ T. The group B has an orbit of cardinality 4 on the non-exceptional section. A similar argument shows that G is conjugate to a group of automorphisms of F_{2}, F_{0}, or ℙ^{2}.

B ≅ O. The group B has an orbit of cardinality 6. As in the previous case we show that G is conjugate to a group of automorphisms of ℙ^{2}, or F_{0}, or F_{2}, or F_{3}.

B ≅ I. The group B has an orbit of cardinality 12. We obtain that G is conjugate to a group of automorphisms of ℙ^{2} or of F_{n}, where n = 0, 2, 3, 4, 5, 6.

• d = 9.

In this case S = ℙ^{2} and G is a finite subgroup of PGL(3). The methods of the representation theory allows us to classify them up to conjugacy in the group PGL(3). However, some of non-conjugate groups can be still conjugate inside the Cremona group.

For example all cyclic subgroups of PGL(3) of the same order n are conjugate in Cr(2). Any element g of order n in PGL(3) is conjugate to a transformation g given in affine coordinates by the formula (x, y) ↦ (ϵ_{n}x, ϵ_{n}^{a}y). Let T ∈ dJ(2) be given by the formula (x, y) ↦ (x, x^{a}/y). Let g′ : (x, y) ↦ (ϵ_{n}^{−1}x, y). We have

g′ ∘ T ∘ g : (x, y) ↦ (ϵ_{n}x, ϵ_{n}^{a}y) ↦ (ϵ_{n}x, x^{a}/y) ↦ (x, x^{a}/y) = T.

This shows that g′ and g are conjugate.

We do not know whether any two isomorphic non-conjugate subgroups of PGL(3) are conjugate in Cr(2).

9 What is left?

Here we list some problems which have not been yet resolved.

• Find the conjugacy classes in Cr(2) of subgroups of PGL(3). For example, there are two non-conjugate subgroups of PGL(3) isomorphic to A_{5} or A_{6} which differ by an outer automorphism of the groups. Are they conjugate in Cr(2)?

• Find the finer classification of the conjugacy classes of de Jonquiéres groups.

We already know that the number of singular fibres in a minimal conic bundle G-surface is an invariant. Even more, the projective equivalence class of the corresponding k points on the base of the conic fibration is an invariant. Are there other invariants? In the case when G_{K} ≅ 2, we know that the quotient of the conic bundle by the involution generating G_{K} is a minimal rule surface F_{e}. Is the number e a new invariant?

• Give a finer geometric description of the algebraic variety parametrizing conjugacy classes.

Even in the case of Del Pezzo surfaces we give only normal forms. What is precisely the moduli space of Del Pezzo surfaces with a fixed isomorphism class of a minimal automorphism group?

We know that conic bundles (S, G) with k ≥ 8 singular fibres are superrigid, so any finite subgroup G′ of Cr(2) conjugate to G is realized as an automorphism group of a conic bundle obtained from S by a composition of elementary transformations with G-invariant centers. If S is not exceptional and G ≅ 2.P, then the invariant of the conjugacy class is the hyperelliptic curve of fixed points of the central involution. If G ≅ 2^{2}.P, then we have three commuting involutions and their curves of fixed points are the invariants of the conjugacy class. Do they determine the conjugacy class?

When k = 6, 7 we do not know whether (S, G) is birationally isomorphic to (S′, G), where S′ is a Del Pezzo surface. This is true when k ∈ {0, 1, 2, 3, 5} or k = 4 and S is not exceptional.

• Find more explicit description of groups G as subgroups of Cr(2).

This has been done in the case of abelian groups in [6]. For example one may ask to reprove and revise Autonne’s classification of groups whose elements are quadratic transformations [3]. An example of such non-cyclic group is the group of automorphisms S_{5} of a Del Pezzo surface of degree 5.

• Finish the classical work on the birational classification of rational cyclic planes z^{n} = f(x, y).

More precisely, the quotient S/G of a rational surface S by a cyclic group of automorphisms defines a cyclic extension of the fields of rational functions. Thus there exists a rational function R(x, y) such that there exists an isomorphism of fields ℂ(x, y)( \sqrt[n]{R(x, y)}) ≅ ℂ(x, y), where n is the order of G. Obviously we may assume that R(x, y) is a polynomial f(x, y), hence we obtain an affine model of S in the form z^{n} = f(x, y). A birational isomorphism of G-surfaces replaces the branch curve f(x, y) = 0 by a Cremona equivalent curve g(x, y). The problem is to describe the Cremona equivalence classes of the branch curves which define rational cyclic planes.

For example, when (S, G) is birationally equivalent to (ℙ^{2}, G), we may take f(x, y) = x since all cyclic groups of given order are conjugate in Cr(2). When n = 2, the problem was solved by M. Noether [46] and later G. Castelnuovo and F. Enriques [16] had realized that the classification follows from Bertini’s classification of involutions in Cr(2). When n is prime the problem was studied by A. Bottari [11]. We refer for modern work on this problem to [12], [13].

• Extend the classification to the case of non-algebraically closed fields, e.g. ℚ, and algebraically closed fields of positive characteristic.

Note that there are more automorphism groups in the latter case. For example, the Fermat cubic surface T_{0}^{3} + T_{1}^{3} + T_{2}^{3} + T_{3}^{3} = 0 over a field of characteristic 2 has the automorphism group isomorphic to U(4, 𝔽_{4}), which is a subgroup of index 2 of the Weyl group W (E_{6}).

10 Tables

In the following tables we give the tables of conjugacy classes of subgroups in Cr(2) which are realized as minimal automorphism groups of Del Pezzo surfaces of degree d ≤ 5 and not conjugate to subgroups of automorphisms of minimal rational surfaces or conic bundles. The information about other groups can be found in the corresponding sections of the paper. The tables contain the order of a group G, its structure, the type of a surface on which the group is realized as a minimal group of automorphisms, the equation of the surface, and the number of conjugacy classes, if finite, or the dimension of the variety parametrizing the conjugacy classes.

Order  Type Surface  Equation Conjugacy
2  A_{1}^{7} DP_{2}  XIII  ∞^{6}
2  A_{1}^{8} DP_{1}  XXII
3  3A_{2} DP_{3}  I,III,IV  ∞^{1}
3  4A_{2} DP_{1}  XVIII  ∞^{3}
4  2A_{3} + A_{1} DP_{2}  II,III,V  ∞^{1}
4  2D_{4}(a_{1}) DP_{1} I,VI, X,XVII,XX  ∞^{5}
5  2A_{4} DP_{1}  XIV  ∞^{2}
6  E_{6}(a_{2}) DP_{3}  I,VI  ∞^{1}
6  A_{5} + A_{1} DP_{3}  I, III, IV  ∞^{1}
6  E_{7}(a_{4}) DP_{2}  XI  ∞^{1}
6  A_{5} + A_{2} DP_{2}  VIII  ∞^{1}
6  D_{6}(a_{2}) + A_{1} DP_{2}  II,III,IV,IX  ∞^{1}
6  A_{5} + A_{2} + A_{1} DP_{1}  II,VIII,XIII  ∞^{2}
6  E_{6}(a_{2}) + A_{2} DP_{1}  II,XII  ∞^{2}
6  E_{8}(a_{8}) DP_{1}  XVIII  ∞^{3}
6  2D_{4} DP_{1}  VII,XI  ∞^{1}
6  E_{7}(a_{4}) + A_{1} DP_{1}  II,VIII,XIX  ∞^{4}
8  D_{5} DP_{4}  (40)  1
8  D_{8}(a_{3}) DP_{1}  X  1
9  E_{6}(a_{1}) DP_{3}  I  1
10  E_{8}(a_{6}) DP_{1}  IV,IX,XIV  ∞^{2}
12  E_{6} DP_{3}  III  1
12  E_{7}(a_{2}) DP_{2}  III  1
12  E_{8}(a_{3}) DP_{1}  I,V  ∞^{2}
14  E_{7}(a_{1}) DP_{2}  I  1
15  E_{8}(a_{5}) DP_{1}  IV  1
18  E_{7} DP_{2}  VI  1
20  E_{8}(a_{2}) DP_{1}  IX  1
24  E_{8}(a_{1}) DP_{1}  I  1
30  E_{8} DP_{1}  IV  1

Table 9. Cyclic subgroups

Order Structure Surface  Equation Conjugacy classes
4  2^{2} DP_{4}  ∞^{2}
4  2^{2} DP_{2}  XII  ∞^{5}
4  2^{2} DP_{1}  XXI  ∞^{5}
4  2^{2} DP_{1} V, VI,X,XV,XVII  ∞^{3}
8  2 × 4 DP_{4}  (39)  ∞^{1}
8  2 × 4 DP_{2}  V  2 × ∞^{1}
8  2 × 4 DP_{2}  I-V  ∞^{1}
8  2 × 4 DP_{2}  VII  ∞^{2}
8  2 × 4 DP_{1}  VIII,XVI  ∞^{2}
8  2^{3} DP_{4}  ∞^{2}
8  2^{3} DP_{2}  I-V,X  ∞^{3}
9  3^{2} DP_{3}  I  1
9  3^{2} DP_{3}  I,IV  2 × ∞^{1}
9  3^{2} DP_{3}  III  1
9  3^{2} DP_{1}  I,II,III  ∞^{1}
12  2 × 6 DP_{4}  (40)  1
12  2 × 6 DP_{2}  III, VIII  ∞^{1}
12  2 × 6 DP_{1}  II,VIII,XIII  ∞^{2}
12  2 × 6 DP_{1}  III,XII  ∞^{2}
12  2 × 6 DP_{1}  II,VII  ∞^{1}
16  2^{4} DP_{4}  ∞^{2}
16  2^{2} × 4 DP_{2}  II,III,V  ∞^{1}
16  4^{2} DP_{2}  II  1
16  2 × 8 DP_{2}  II  1
18  3 × 6 DP_{3}  I  1
18  3 × 6 DP_{1}  III  ∞^{1}
18  3 × 6 DP_{1}  II  1
24  2 × 12 DP_{1}  VIII  ∞^{1}
24  2 × 12 DP_{2}  III  1
27  3^{3} DP_{3}  I  1
32  2 × 4^{2} DP_{2}  II  1
36  6^{2} DP_{1}  II  1

Table 10. Abelian non-cyclic subgroups

Order Structure Surface  Equation Conjugacy classes
6  D_{6} DP_{3} III,IV,VIII,XI  ∞^{2}
6  D_{6} DP_{3}  I  1
8  D_{8} DP_{4}  (39)  ∞^{1}
8  D_{8} DP_{2} II,III,V,VII  ∞^{2}
8  D_{8} DP_{1}  I, XVI  ∞^{3}
8  Q_{8} DP_{1} II,V,VI,XIV  ∞^{2}
8  3 × D_{8} DP_{1}  I, XVI  ∞^{3}
8  3 × Q_{8} DP_{1}  II,III  ∞^{1}
12  2 × D_{6} DP_{3}  I,VI  ∞^{1}
12  T DP_{2}  II  1
12  2 × D_{6} DP_{2}  I,II,IV,IX  ∞^{2}
12  2 × D_{6} DP_{2}  II  1
12  2 × D_{6} DP_{1}  I,II,III,VI  ∞^{1}
16  2 × D_{8} DP_{2} II,III,V,VII  ∞^{2}
16  D_{16} DP_{1}  I,IX  ∞^{1}
18  3 × D_{6} DP_{3}  III  1
18  3 × D_{6} DP_{3}  I,IV  2 × ∞^{1}
18  3 × D_{6} DP_{1}  I,II,III  ∞^{1}
24  \overline{T} DP_{1}  I,V  ∞^{1}
24  2 × T DP_{4}  (41)  1
24  S_{4} DP_{3}  I  3
24  S_{4} DP_{3}  II  1
24  S_{4} DP_{2}  II  1
24  S_{4} DP_{2}  I, II,IV  ∞^{1}
24  2 × T DP_{2}  II  1
24  3 × Q_{8} DP_{1}  I  1
24  3 × Q_{8} DP_{1}  II  1
24  3 × D_{8} DP_{1}  I  ∞^{2}
36  6 × D_{6} DP_{1}  I,II,III  ∞^{1}
48  2 × O DP_{2}  I,II,III,V  ∞^{1}
48  2 × \overline{T} DP_{1}  I  1
72  3 × \overline{T} DP_{1}  I  1

Table 11. Products of cyclic groups and polyhedral or binary polyhedral non-cyclic group

Order Structure Surface  Equation Conjugacy classes
8  D_{8} DP_{4}  (39)  ∞^{1}
8  D_{8} DP_{2} II,III,IV,V,VII  ∞^{2}
8  Q_{8} DP_{1}  I,V,XIV  ∞^{2}
8  D_{8} DP_{1}  I,V,XVI  ∞^{2}
16  L_{16} DP_{4}  (39)  2 × ∞^{1}
16  2 × D_{8} DP_{2} II,III,IV,V,VII  1
16  2 × D_{8} DP_{2}  I  1
16  AS_{16} DP_{2}  II,III,V  2 × ∞^{1}
16  M_{16} DP_{2}  II  1
16  D_{16} DP_{1}  I,IX  ∞^{1}
32  2^{2} : 8 DP_{4}  (40)  1
32  2^{4} : 2 DP_{4}  (39)  ∞^{1}
32  D_{8} : 4 DP_{2}  II  1
32  2 × AS_{16} DP_{2}  III,V  3 × ∞^{1}
32  2 × M_{16} DP_{2}  II  1
64  2^{4} : 4 DP_{4}  (40)  1
64 2 × (D_{8} : 4) DP_{2}  II  1
27  H_{3}(3) DP_{3}  I,III,IV  ∞^{1}
81  3^{3} : 3 DP_{3}  I  1

Table 12. Non-abelian p-groups

Order Structure Surface Equation Conjugacy classes
18  3^{2} : 2 DP_{3}  I  2
24  2D_{12} DP_{1}  II,VI  ∞^{1}
24  D_{8} : 3 DP_{2}  III  1
36  3^{2} : 2^{2} DP_{3}  I  1
42  2 × (7 : 3) DP_{1}  II  1
48  2^{4} : 3 DP_{4}  (40)  1
48  2 × D_{8} : 3 DP_{2}  III  1
48  \overline{T} : 2 DP_{1}  I  1
48  4^{2} : 3 DP_{2}  II  1
54  H_{3}(3) : 2 DP_{3}  IV  ∞^{1}
54  3^{3} : 2 DP_{3}  I  2
60  A_{5} DP_{5}  1
72  3 × 2D_{12} DP_{1}  II  1
80  2^{4} : 5 DP_{4}  (41)  1
96  2^{4} : S_{3} DP_{4}  (40)  1
96  4^{2} : S_{3} DP_{2}  I  2
96  2 × (4^{2} : 3) DP_{2}  II  1
108  3^{3} : 4 DP_{3}  I  2
108  3^{3} : 2^{2} DP_{3}  I  2
120  S_{5} DP_{5}  1
120  S_{5} DP_{3}  II  1
144  3 × (\overline{T} : 2) DP_{1}  I  1
160  2^{4} : D_{10} DP_{4}  (41)  1
162  3^{3} : S_{3} DP_{3}  I  2
168  L_{2}(7) DP_{2}  I  1
192 2 × (4^{2} : S_{3}) DP_{3}  I  1
216  3^{3} : D_{8} DP_{3}  I  2
336  2 × L_{2}(7) DP_{2}  I  1
648  3^{3} : S_{4} DP_{3}  I  2

Table 13. Other groups

References

1. A. Adem, J. Milgram, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, 309, Springer-Verlag, Berlin, 1994.

2. M. Alberich-Carramiana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769. Springer-Verlag, Berlin, 2002. xvi+257 pp.

3. L. Autonne, Recherches sur les groupes d’ordre fini contenus dans le groupe Cremona, Prem. Mm. Gnralits et groupes quadratiques, J. Math. Pures et Appl., (4) 1 (1885), 431–454.

4. S. Bannai, H. Tokunaga, A note on embeddings of S_{4} and A_{5} into the Cremona group and versal Galois covers, math.AG/0510503, 12 pp.

5. L. Bayle, A. Beauville, Birational involutions of P^{2}, Asian J. Math. 4 (2000), 11–17.

6. J. Blanc, Finite abelian subgroups of the Cremona group of the plane, Thesis, Univ. of Geneva, 2006.

7. A. Beauville, J. Blanc, On Cremona transformations of prime order, C. R. Math. Acad. Sci. Paris 339 (2004), 257–259.

8. A. Beauville, p-elementary subgroups of the Cremona group, math. AG/0502123, J. of Algebra, to appear.

9. E. Bertini, Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat. Pura Appl. (2) 8 (1877), 254–287.

10. H. Blichfeldt, Finite collineation groups, with an introduction to the theory of operators and substitution groups, Univ. of Chicago Press, Chicago, 1917.

11. A. Bottari, Sulla razionalità dei piani multipli \{x, y, \sqrt[n]{F (x, y)}\}, Annali di Mat. Pura Appl. (3) 2 (1899), 277–296.

12. A. Calabri, Sulle razionalità dei piani doppi e tripli cyclici, Ph. D. thesis, Univ. di Roma “La Sapienza”, 1999.

13. A. Calabri, On rational and ruled double planes, Annali. di Mat. Pura Appl. (4) 181 (2002), 365–387.

14. R. Carter, Conjugacy classes in the Weyl group. in “ Seminar on Algebraic Groups and Related Finite Groups”, The Institute for Advanced Study, Princeton, N.J., 1968/69, pp. 297–318, Springer, Berlin.

15. G. Castelnuovo, Sulle razionalità delle involutioni piani, Math. Ann, 44 (1894), 125–155.

16. G. Castelnuovo, F. Enriques, Sulle condizioni di razionalità dei piani doppia, Rend. Circ. Mat. di Palermo, 14 (1900), 290–302.

17. A. Coble, Algebraic geometry and theta functions (reprint of the 1929 edition), A. M. S. Coll. Publ., v. 10. A. M. S., Providence, R.I., 1982. MR0733252 (84m:14001)

18. J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson, Atlas of finite groups, Oxford Univ. Press, Eynsham, 1985.

19. J. Conway, D. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003.

20. J. Coolidge, A treatise on algebraic plane curves, Dover Publ. New York. 1959.

21. A. Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), 223–254.

22. T. de Fernex, On planar Cremona maps of prime order, Nagoya Math. J. 174 (2004), 1–28.

23. T. de Fernex, L. Ein, Resolution of indeterminacy of pairs, in “Algebraic geometry”, pp. 165–177, de Gruyter, Berlin, 2002.

24. M. Demazure, Surfaces de Del Pezzo, I-V, in “Séminaire sur les Singularités des Surfaces”, Ed. by M. Demazure, H. Pinkham and B. Teissier. Lecture Notes in Mathematics, 777. Springer, Berlin, 1980, pp. 21–69.

25. I. Dolgachev, Weyl groups and Cremona transformations. in “Singularities, Part 1 (Arcata, Calif., 1981)”, 283–294, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.

26. I. Dolgachev, Topics in classical algebraic geometry, Part I, manuscript in preparations, see www.math.lsa.umich.edu/ idolga/lecturenotes.html.

27. P. Du Val, On the Kantor group of a set of points in a plane, Proc. London Math. Soc. 42 (1936), 18–51.

28. F. Enriques, Sulle irrazionalita da cui puo farsi dipendere la resoluzione d’un’ equazione algebrica f(x, y, z) = 0 con funzioni razionali di due parametri, Math. Ann. 49 (1897), 1–23.

29. C. Geiser, Über zwei geometrische Probleme, J. Reine Angew. Math. 67 (1867), 78–89.

30. D. Gorenstein, Finite groups, Chelsea Publ. Co., New York, 1980.

31. É. Goursat, Sur les substitutions orthogonales et les divisions rgulires de l’espace, Ann. de l’cole Norm. Sup. (3) 6 (1889), 9–102.

32. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

33. T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), 651– 677.

34. H. Hudson, Cremona transformations in plane and space, Cambridge Univ. Press. 1927.

35. V. A. Iskovskih, Rational surfaces with a pencil of rational curves (Russian) Mat. Sb. (N.S.) 74 (1967), 608–638.

36. V. A. Iskovskikh, Rational surfaces with a pencil of rational curves with positive square of the canonical class, Math. USSR Sbornik, 12 (1970), 93–117.

37. V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 19–43,

38. V. A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of view of Mori theory (Russian) Uspekhi Mat. Nauk 51 (1996), 3–72; translation in Russian Math. Surveys 51 (1996), 585–652.

39. V. A. Iskovskikh, Two nonconjugate embeddings of the group S_{3} × Z_{2} into the Cremona group (Russian) Tr. Mat. Inst. Steklova 241 (2003), Teor. Chisel, Algebra i Algebr. Geom., 105–109; translation in Proc. Steklov Inst. Math. 241 (2003), 93–97.

40. V. A. Iskovskikh, Two non-conjugate embeddings of S_{3} × ℤ_{2} into the Cremona group II, math.AG/0508484.

41. S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin. Mayer & Mller. 111 S. gr. 8^{∘}. 1895.

42. J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.

43. N. Lemire, V. Popov, Z. Reichstein, Cayley groups, math.AG/0409004

44. Yu. I. Manin, Rational surfaces over perfect fields. II (Russian) Mat. Sb. (N.S.) 72 (1967), 161–192.

45. Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic Translated from Russian by M. Hazewinkel. North-Holland Mathematical Library, Vol. 4. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974.

46. M. Noether, Über die ein-zweideutigen Ebenentransformationen, Sitzungberichte der physic-medizin, Soc. zu Erlangen, 1878.

47. B. Segre, The non-singular cubic surface, Oxford Univ. Press. Oxford. 1942.

48. T. Springer, Invariant theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin-New York, 1977.

49. A. Wiman, Zur Theorie endlichen Gruppen von birationalen Transformationen in der Ebene, Math. Ann. 48 (1896), 195–240.

50. H. Zassenhaus, The theory of groups, Chelsea Publ. New York, 1949.

51. D.-Q. Zhang, Automorphisms of finite order on rational surfaces. With an appendix by I. Dolgachev, J. Algebra 238 (2001), 560–589.

Finite subgroups of the plane Cremona group  543