Speakers:  Vadim Vologodsky (MIT)

Title: Semistable models for Severi-Brauer varieties.

Abstract: Let R be a hensilian discrete  valuation domain (e.g., R=K[[t]]) with fraction field F, and let  X be a Severi-Brauer variety over F. We say that X is tame if it has a point over an unramified extension of F (this is always the case if the residue field of R is perfect).

We prove that X has a proper semistable model over R if and only if X is tame. 

The geometric special fiber of the semistable model we construct is a certain quiver Grassmannian. The proof makes use of a description of hereditary R-orders (i.e. orders whose Jacobson radical is a projective module over the order) in the algebra of matrices over F obtained by Brumer.

   The talk is based on a joint work in progress with Alexei Kubanov and Constantin Shramov.