Abstracts - Spring 2013 

Abstracts Spring 2013

Speaker:  Ben Antieau

Title:  Derived categories of principal homogeneous spaces of abelian varieties

Abstract: I will give a general introduction to derived equivalences between smooth projective varieties. Then, I will describe recent work on the problem of determining when when two principal homogeneous spaces for an abelian variety are derived equivalent, culminating in a precise description of when two smooth projective curves over a field of characteristic zero have equivalent bounded derived categories.

Speaker: Izzet Coskun

Title: The birational geometry of the Hilbert scheme of points on P^2 and Bridgeland stability

Abstract:  In this talk, I will describe joint work with Daniele Arcara, Aaron Bertram and Jack Huizenga on running the Minimal Model Program (MMP) for the Hilbert scheme?of points on P^2. We study

the stable base locus decomposition of the effective cone of the Hilbert?scheme. We find a correspondence between the decomposition of the effective cone into the Mori chambers and the decomposition of the stability manifold into Bridgeland chambers.

Speaker:  Karl Schwede

Title:  F-singularities in families

Abstract:  F-singularities are classes of singularities defined by the behavior of Frobenius. A prominent tool for measuing these singularities is the test ideal, a characteristic p > 0 analog of the multiplier ideal. Recently, there has been interest in applying the methods of F-singularities to a number of geometric problems in positive characteristic. However, one gap in

the theory has been the behavior of F-singularities in families. For example generic restriction theorems for test ideals have been lacking.

In this talk, I will discuss recent joint work with Zsolt Patakfalvi and Wenliang Zhang on the behavior of F-singularities and test ideals in families. I will first define the relevant terms and explain why you these definitions and methods are useful (as a replacement for Kodaira vanishing in characterisic p > 0).  We will then obtain generic (and non-generic) restrictions theorems for test ideals. Some global geometric consequences will also be discussed if there is sufficient time.

Speaker:  Anastasia Stavrova

Title:  Isotropic reductive groups and non-stable K-theory

Abstract: A semisimple (more generally, reductive) group scheme is called isotropic, if it contains a proper parabolic subgroup. In particular, the special orthogonal group of a non-degenerate quadratic form is isotropic if and only if the form itself is isotropic. To any isotropic reductive group G over a commutative ring R one associates a group-valued functor K_1^G on the category of commutative R-algebras, which is an analog of the non-stable K_1-functor GL_n/E_n. When R is a field, K_1^G coincides on smooth algebras with the group of A^1-connected components of G in the sense of Morel-Voevodsky. We will discuss various properties of these functors, and connections with classification of principal G-bundles.