Algebraic Geometry Seminar

Multiplicity-free products of Schubert divisors and an application to canonical dimension.

Speaker: Rostislav Devyatov

Location: Warren Weaver Hall 317

Date: Tuesday, February 19, 2019, 3:30 p.m.


Let G/B be a flag variety over C, where G is a simple algebraic group
with a simply laced Dynkin diagram, and B is a Borel subgroup. The
Bruhat decomposition of G defines subvarieties of G/B called Schubert
subvarieties. The codimension 1 Schubert subvarieties are called
Schubert divisors. The Chow ring of G/B is generated as an abelian
group by the classes of all Schubert varieties, and is "almost"
generated as a ring by the classes of Schubert divisors. More
precisely, an integer multiple of each element of G/B can be written
as a polynomial in Schubert divisors with integer coefficients. In
particular, each product of Schubert divisors is a linear combination
of Schubert varieties with integer coefficients.

In the first part of my talk I am going to speak about the
coefficients of these linear combinations. In particular, I am going
to explain how to check if a coefficient of such a linear combination
is nonzero and give an idea how to check if such a coefficient equals
1. And in the second part of my talk, I will say something about an
application of my result, namely, how it makes it possible estimate
so-called canonical dimension of flag varieties and groups over
non-algebraically-closed fields.