WAGS Fall 2009

WAGS / Fall 2009

Western
Algebraic
Geometry
Seminar

A special occurrence
in honor of David Gieseker

Abstracts

University of California
Los Angeles
24-25 October 2009

Topology of Particle Collisions
Satyan Devadoss, MSRI/Williams
Our story is motivated by the configuration space of particles on spheres. In the 1970s, Deligne and Mumford constructed a way to keep track of particle collisions in this space using Geometric Invariant Theory. In the 1980s, this (compactified) moduli space was remarkably used by Gromov and Witten as invariants arising from string field theory. In the 1990s, Kontsevich and Fukaya generalized these ideas when studying deformation quantization to include particles collisions on surfaces with boundary. This talk, using visual brushstrokes, focuses on the topology of real points of particle collisions. These real analogs can be understood from several viewpoints: from tiling of convex polytopes, to blowups of hyperplane arrangements, to underlying operad structures, to spaces of phylogenetic trees, and to open-closed string field theory.

Our story is motivated by the configuration space of particles on spheres. In the 1970s, Deligne and Mumford constructed a way to keep track of particle collisions in this space using Geometric Invariant Theory. In the 1980s, this (compactified) moduli space was remarkably used by Gromov and Witten as invariants arising from string field theory. In the 1990s, Kontsevich and Fukaya generalized these ideas when studying deformation quantization to include particles collisions on surfaces with boundary.

This talk, using visual brushstrokes, focuses on the topology of real points of particle collisions. These real analogs can be understood from several viewpoints: from tiling of convex polytopes, to blowups of hyperplane arrangements, to underlying operad structures, to spaces of phylogenetic trees, and to open-closed string field theory.

Birational geometry of holomorphic symplectic varieties
Brendan Hassett, Rice
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also present evidence supporting our conjectures in specific cases. (joint with Y. Tschinkel)

Birational geometry of holomorphic symplectic varieties

Brendan Hassett, Rice

We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also present evidence supporting our conjectures in specific cases. (joint with Y. Tschinkel)

BGG Correspondence, Cohomology of Compact Kahler Manifolds, and Applications
Robert Lazarsfeld, Michigan
Let X be a compact Kahler manifold of dimension d, and denote by E the exterior algebra on H^1(X, O_X). One can view P_X = H^(X, O_X) and Q_X = H^(X, \omega_X) as modules over E. P_X does not seem to have any simple algebraic structure as an E-module, but when X has large Albanese dimension it turns out that a body of work involving generic vanishing theorems implies that Q_X is surprisingly well behaved. The connection proceeds via the BGG correspondence, which relates modules over an exterior algebra to linear complexes over the symmetric algebra. We will discuss the the algebraic properties of Q_X, and show that this E-module contains a surprising amount of information about the geometry of X. We then use a vector bundle arising from the BGG correspondence to establish, under mild additional hypotheses, a number of inequalities on Hodge numbers and the holomorphic Euler characteristic of X. This is joint work with Mihnea Popa.

BGG Correspondence, Cohomology of Compact Kahler Manifolds, and Applications

Robert Lazarsfeld, Michigan

Let X be a compact Kahler manifold of dimension d, and denote by E the exterior algebra on H^1(X, O_X). One can view

P_X = H^*(X, O_X) and Q_X = H^*(X, \omega_X)

as modules over E. P_X does not seem to have any simple algebraic structure as an E-module, but when X has large Albanese dimension it turns out that a body of work involving generic vanishing theorems implies that Q_X is surprisingly well behaved. The connection proceeds via the BGG correspondence, which relates modules over an exterior algebra to linear complexes over the symmetric algebra. We will discuss the the algebraic properties of Q_X, and show that this E-module contains a surprising amount of information about the geometry of X. We then use a vector bundle arising from the BGG correspondence to establish, under mild additional hypotheses, a number of inequalities on Hodge numbers and the holomorphic Euler characteristic of X. This is joint work with Mihnea Popa.

Degeneration of moduli spaces, following Gieseker
Jun Li, Stanford
I will talk about the degeneration method in moduli problems inspired by Gieseker. The primary examples are moduli of stable maps and stable sheaves. Recent applications will be outlined.

Categorical invariants for cubics
Emanuele Macri, Utah
It is a theorem of Clemens-Griffiths and Tyurin that a cubic threefold is determined by its intermediate Jacobian. We discuss a categorical version of this theorem, namely, that a cubic threefold is determined by a semi-orthogonal component of its derived category. Our approach is to construct an interesting Bridgeland stability condition on this component and recover the Fano surface of lines as a moduli space of stable objects. If time permits, we will also discuss some generalization to the case of cubic fourfolds. This is joint work with P. Stellari, and partly with M. Bernardara and S. Mehrotra.

Categorical invariants for cubics

Emanuele Macri, Utah

It is a theorem of Clemens-Griffiths and Tyurin that a cubic threefold is determined by its intermediate Jacobian. We discuss a categorical version of this theorem, namely, that a cubic threefold is determined by a semi-orthogonal component of its derived category. Our approach is to construct an interesting Bridgeland stability condition on this component and recover the Fano surface of lines as a moduli space of stable objects. If time permits, we will also discuss some generalization to the case of cubic fourfolds. This is joint work with P. Stellari, and partly with M. Bernardara and S. Mehrotra.

Vector bundles with sections
Brian Osserman, UC Davis
Classical Brill-Noether theory studies, for given g, r, d, the space of line bundles of degree d with r+1 global sections on a curve of genus g. After reviewing the main results in this theory, and the role of degeneration techniques in proving them, we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open. Focusing on the case of rank 2, we will discuss the role of spaces with fixed determinant, and how the dimension theory of Artin stacks may be useful in applying degeneration techniques.

Vector bundles with sections

Brian Osserman, UC Davis

Classical Brill-Noether theory studies, for given g, r, d, the space of line bundles of degree d with r+1 global sections on a curve of genus g. After reviewing the main results in this theory, and the role of degeneration techniques in proving them, we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open. Focusing on the case of rank 2, we will discuss the role of spaces with fixed determinant, and how the dimension theory of Artin stacks may be useful in applying degeneration techniques.

The boundary of the p-rank strata of the moduli space of hyperelliptic curves
Rachel Pries, Colorado State
In characteristic p, the moduli space of curves can be stratified by an arithmetic invariant, the p-rank, which measures the number of p-torsion points on the Jacobian. I will talk about the geometric structure of the boundary of the p-rank strata, with an emphasis on new results in the hyperelliptic case. These results yield many applications about l-adic monodromy and Jacobians of curves with given p-rank. This is joint work with Jeff Achter.

The boundary of the p-rank strata of the moduli space of hyperelliptic curves

Rachel Pries, Colorado State

In characteristic p, the moduli space of curves can be stratified by an arithmetic invariant, the p-rank, which measures the number of p-torsion points on the Jacobian. I will talk about the geometric structure of the boundary of the p-rank strata, with an emphasis on new results in the hyperelliptic case. These results yield many applications about l-adic monodromy and Jacobians of curves with given p-rank. This is joint work with Jeff Achter.

Department of Mathematics
University of California

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