Talk-abstracts

Angela Gibney

Title: Vector bundles on moduli of curves from vertex operator algebras

Abstract: Vertex operator algebras V are certain infinite dimensional vector spaces with extra structure, whose representations may naturally be associated with a pointed algebraic curve of genus g. This talk concerns algebraic structures on the moduli space of Deligne-Mumford stable n-pointed curves of genus g arising from collections of n fixed modules V-modules.   As will be discussed, for nice enough VOAs V, and V-modules,  through this construction one obtains a vector bundle. Until recently, the main examples for all genera were constructed from representations of affine VOAs at positive integer levels, and strongly rational VOAs in genus zero. Building on this work, with my coauthors and students, we have extended this theory, and a bigger picture is emerging. In this overview, I will describe the motivation behind this work, the current state of our knowledge, and some of what we hope to prove.

Tony Várilly-Alvarado

Title: Geometric realizations of Brauer classes on K3 surfaces via Hyperkaehler Geometry

Abstract: Brauer groups of K3 surfaces have amply documented sightings in several subfields of algebraic and arithmetic geometry.  They bear on the rationality problem of fourfolds; they mediate the arithmetic of surfaces over nonclosed fields; they play a starring role in the geometry of birational maps between hyperkaehler varieties; they keep track of the data of Hodge isogenies between K3 surfaces; on and on.  Geometric realizations of Brauer classes on K3 surfaces, as etale projective bundles, help us understand many of these applications.  I will discuss joint work with Jack Petok and Sarah Frei on constructing these geometric realizations. Our work builds on earlier efforts and predictions of Hassett and Tschinkel, as well as on recent results of van Geemen and Kaputska, who show that some 2-torsion Brauer classes on K3 surfaces have realizations as the exceptional locus of a divisorial contraction on a hyperkaehler fourfold.

Sebastián Olano

Title: Partial Rational Smoothness

Abstract: A complex variety Z is called rationally smooth, or a rational homology manifold, if, at every point, the local cohomologies $H^k_{\{x\}}(Z)$ are that of a sphere of dimension $2\dim{Z}$. While smooth varieties are rationally smooth, there are several examples of singular varieties that satisfy this condition. These varieties exhibit interesting geometric properties, including Poincaré duality. We study a natural weakening of this notion, which we call partial rational smoothness. This notion captures the difference between higher Du Bois and higher rational singularities, two classes of singularities that have recently attracted significant interest.

Laure Flapan

Title: Noether-Lefschetz divisors and moduli spaces of K3 surfaces

Abstract: Noether-Lefschetz divisors on moduli spaces of K3 surfaces parametrize K3 surfaces with Picard rank bigger than 1. We discuss the relationship between these divisors and certain modular forms and describe how these modular forms can be used to understand the geometry of these moduli spaces.

Xiaolei Zhao

Title: Noncommutative abelian surfaces and Kummer-type hyperkähler varieties

Abstract: Polarised abelian surfaces vary in three-dimensional families. In contrast, the derived category of an abelian surface A has a six-dimensional space of deformations; moreover, based on general principles, one should expect to get "algebraic families" of their categories over four-dimensional bases. Kummer-type hyperkähler varieties (KHK) are hyperkähler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised KHKs have four-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarieties. I present a construction that addresses both issues. We construct four-dimensional families of categories that are deformations of Db (A) over an algebraic space. Moreover, each category admits a Bridgeland stability conditions, and from the associated moduli spaces of stable objects one can obtain every general polarised KHK, for every possible polarisation type of KHKs. Our categories are obtained from Z/2-actions on derived categories of K3 surfaces. This is based on joint work with Arend Bayer, Alex Perry and Laura Pertusi.

Maria Fox

Title: Ekedahl-Oort and Newton Stratifications

Abstract: We will discuss two important invariants of abelian varieties in characteristic p: the p-torsion subgroup and the p-divisible group. Given a moduli space $\mathcal{M}$ of abelian varieties in characteristic p, these two invariants can each be used to create a stratification of $\mathcal{M}$. We will see in several concrete examples that these two stratifications are very different, reflecting the fact that these two invariants capture very different attributes of the abelian varieties.  The new result in this talk is joint with E. Anne, D. Bhamidipati, H. Goodson, S. Groen, and S. Nair.

Sebastian Casalaina-Martin

Title: Moduli spaces of cubic hypersurfaces

Abstract: In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces.  A focus will be on explaining why two natural models of the moduli space of cubic surfaces are not isomorphic, or even K-equivalent.  I will also discuss a related moduli space, the moduli space of cubic surfaces with a marked line.