## FRG 2012 Workshop

Abstracts

Integral Hodge classes on fourfolds with quadric fibration, Zhiyuan Li (Stanford University)
It is well known that the Hodge conjecture (HC) holds for Hodge classes on uniruled fourfolds. It remains open when HC holds for integral Hodge classes. Due to the work of Voisin, this problem is related to the study of Abel-Jacobi map on threefolds. .

In this talk, we consider the space of sections and certain bisections on threefolds with quadratic surfaces fibration over a curve and we show that the Abel-Jacobi from these spaces to the intermediate Jacobian is dominant with rationally connected fibration. The proof is similar to Hassett and Tschinkel's work of rational points on surfaces over function fields. As an application, we prove that HC holds on degree four integral Hodge classes of families of such threefolds. This is a joint work with Zhiyu Tian.

A classification of Lagrangian planes via Bridgeland stability conditions, Benjamin Bakker (Courant Institute)Recent work of Bayer and Macri has beautifully described the birational geometry of moduli spaces of sheaves on K3 surfaces using Bridgeland stability conditions. In particular, the various birational cones associated to such a moduli space are determined purely in terms of intersection theory, and Bayer, Hassett, and Tschinkel show that the same formula for the Mori cone deforms to all varieties of K3^[n] type. Using these methods, we prove that contractible Lagrangian planes in varieties of K3^[n] type are likewise classified purely by the intersection theoretic properties of the class of the line sweeping them out. We will also discuss the relationship between the geometry of more general extremal contractions and the lattice properties of their extremal rays.

Tameness in arithmetic geometry and existence of slices, Sophie Marques (Courant Institute)Arithmetic applications of Brauer groups on degree 2 K3 surfaces, Sho Tanimoto (Rice University)

Recently Hassett and Varilly showed that transcendental elements of the Brauer group of a general K3 surface can obstruct Weak approximation and Hasse principle. I will report our recent progress on their results. This is joint work with Kelly McKinnie, Justin Sawon, and Anthony Varilly.

Higher rank interpolation problems and the birational geometry of Hilbert schemes of points, Jack Huizenga (University of Illinois at Chicago)Questions like the Nagata conjecture seek to determine when certain zero-dimensional schemes impose independent conditions on sections of a line bundle on a surface. Understanding analogous questions for vector bundles instead of line bundles amounts to studying the birational geometry of Hilbert schemes of points on a surface. In this talk we will discuss how Bridgeland stability allows one to solve interesting higher rank interpolation problems.

Interpolation, birational geometry of moduli spaces of sheaves and Bridgeland stability, Izzet Coskun (University of Illnois at Chicago)In this talk, I will report on joint work with Jack Huizenga on higher rank interpolation problems and the birational geometry of moduli spaces of sheaves on the projective plane. The new perspective offered by Bridgeland stability enables us to solve the higher rank interpolation problem for monomial schemes. Similar techniques allow us to describe the effective cones of all moduli spaces of semi-stable sheaves on the plane.