# Algebraic Geometry Seminar

#### Rationality Questions and Quartic Threefolds

**Speaker:**
Alena Pirutka, Centre de Mathématiques Laurent Schwartz

**Location:**
Warren Weaver Hall 201

**Date:**
Tuesday, October 7, 2014, 3:30 p.m.

**Synopsis:**

In the context of the Lüroth problem, for K a function field of a smooth projective variety X over a field k, one can ask whether K is purely transcendental over k (resp. a subfield of a purely transcendental extension of k, resp. becomes purely transcendental after adding some independent variables), that is, if X is rational (resp. unirational, resp. stably rational). In this talk we will discuss various invariants which allow us to answer these questions for some classes of varieties, more specifically, for quartic threefolds. By a celebrated result of Iskovskikh and Manin, no smooth quartic hypersurface in CP^{4} is rational. Using a specialization method introduced by C. Voisin, as well as a method based on the universal properties of the Chow group of zero-cycles, we will show that many such quartics are not stably rational. This is joint work with J.-L. Colliot-Thélène.