December 7-9, 2018 at University of Illinois at Chicago
Titles and Abstracts:
Friday, December 7, 2018
Arend Bayer: Surfaces from curves via derived categories. 3:00-4:00. Lecture Center D, Room 5
I will explain how the seemingly highly abstract machinery of derived categories can be used to answer fundamental and concrete questions in algebraic geometry. I will give several examples of this philosophy; the one alluded to in the title is due to Soheyla Feyzbakhsh, who showed that a generic K3 surface X can be geometrically reconstructed from any curve in X of minimal possible degree.
Benjamin Schmidt: Derived Categories and the Genus of Space Curves. 4:15-5:15. Lecture Center D, Room 2.
A 19th century problem is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. In this talk I will give an introduction to the topic and explain how it is related to Bridgeland stability. From a modern perspective there is no reason to restrict to the case of projective space, and I will discuss recent progress on the question in the case of abelian threefolds in joint work with Emanuele Macri.
Saturday, December 8, 2018
All talks today will take place in Lecture Center D, Room 4.
Arend Bayer: Stability conditions in families. 9:30-11:00
I will give an overview of the construction of stability conditions in a family of varieties, in current work in preparation with Marti Lahoz, Howard Nuer, Emanuele Macri, Alex Perry and Paolo Stellari, along with existing and potential applications.
Sarah Frei: Moduli spaces of sheaves on K3 surfaces and Galois representations. 11:20-12:00
There have been some interesting results in recent years about derived equivalences of varieties implying equality of their zeta functions. In this talk, I will discuss a new result in this direction for moduli spaces of stable sheaves on K3 surfaces. It turns out that, regardless of derived equivalence, two such moduli spaces of the same dimension have the same zeta function. The way to get to this result is to study the cohomology groups of the moduli spaces as Galois representations.
Daniel Levine: Brill-Noether for Del Pezzo surfaces. 1:30-2:10
Interesting loci in moduli spaces of vector bundles can be constructed by considering sets of bundles where cohomology "jumps." The study of these loci is intimately connected with the cohomology of a general bundle. On the projective plane, G\"ottsche and Hirschowitz showed that a general stable vector bundle has at most one nonzero cohomology group. For Hirzebruch surfaces, this statement is false in general, but Coskun and Huizenga compute the cohomology of a general bundle. In joint work with Shizhuo Zhang, we give an analogous classification for Del Pezzo surfaces of degree at least 5.
Benjamin Sung: Discriminants of Rank Two Sheaves on Surfaces. 2:10-2:50
A major obstacle towards the construction of Bridgeland stability conditions and the classification of stable sheaves is an understanding of discriminants and their bounds. I will highlight an approach towards proving a Bogomolov inequality on singular schemes and present partial results in this direction. In particular, I will demonstrate applications of such techniques towards the proof of a sharp bound on discriminants of rank two sheaves on general type surfaces. This is based on joint work with Benjamin Schmidt.
Jarod Alper: Construction of moduli spaces of sheaves. 3:15-4:15
The construction of moduli spaces is a recurring problem in algebraic geometry. We will outline a general technique to construct moduli schemes parameterizing equivalence classes of objects that may have positive dimensional automorphism groups. This technique, developed in collaboration with Daniel Halpern-Leistner and Jochen Heinloth, offers a different perspective to constructing moduli spaces than the classical approach using geometric invariant theory. We will examine this technique through the lens of one of the most familiar moduli spaces in algebraic geometry, namely the moduli space of semistable vector bundles on a smooth projective curve.
Laure Flapan: Kodaira fibrations and moduli of abelian varieties. 4:30-5:30
A Kodaira fibration is a smooth projective surface fibration such that all fibers are smooth. We investigate the question of possible monodromy groups of such fibrations by translating to questions about complete curves on sub-loci of the moduli space of principally polarized abelian varieties.
Sunday, December 9, 2018
All talks today will take place in Lecture Center D, Room 4.
Jarod Alper: Moduli of objects in an abelian category. 9:00-10:30
Building on the work of Artin and Zhang, we will present a construction of a proper moduli space parameterizing S-equivalences of semistable objects in an abelian category. Gieseker semistability and Bridgeland semistability can both be viewed within this framework.
John Kopper: Stability conditions for restrictions of vector bundles on projective surfaces. 10:45-11:25
I will describe some recent results that use Bridgeland stability conditions to give sufficient criteria for a stable vector bundle on a surface to remain stable when restricted to a curve. I will also discuss some applications of these ideas to higher rank Brill-Noether theory.
Franco Rota: Bridgeland stability and mirror symmetry for orbifold elliptic quotients. 11:25-12:05
Mirror symmetry predicts a relation between the stability manifold of an object X and the moduli space of its mirror. We focus on the case where X is a certain orbifold quotient of an elliptic curve. In this case, we show that the stability manifold is a covering space of the universal unfolding of a simple elliptic singularity, and its geometry is regulated by an extended affine root system. This possibly represents an example of a stability manifold admitting a Frobenius manifold structure.