# 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.

**Synopsis:**

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.