Spring 2025
January 21: organizational meeting (on Zoom)
February 4: Aaron Landesman (Harvard University)
Title: The asymptotic Picard rank conjecture
Abstract: The Picard rank conjecture predicts the vanishing of the rational picard group of the Hurwitz space parameterizing simply branched covers of $\mathbb P^1$ of degree $d$ and genus $g$. In joint work with Ishan Levy, we prove the Picard rank conjecture when $g$ is sufficiently large relative to $d$. The main input is a new result in topology where we prove that the homology of Hurwitz spaces stabilize and compute their dominant stable value.
This is, in some sense, a continuation of my talk in the number theory seminar earlier today, though nothing from that talk will be assumed.
March 4: Peter McDonald (University of Illinois Chicago)
Title: An explicit derived McKay correspondence for some complex reflection groups of rank 2
Abstract: Let G be a finite subgroup of $SL_2(\mathbb{C})$. The classical McKay correspondence draws connections between the representation theory of G, the geometry of the minimal resolution of singularities of $\mathbb{C}^2/G$, and the algebra of the invariant ring $\mathbb{C}[x,y]^G$. Thanks to Kapranov-Vasserot, it can also be interpreted as an equivalence of triangulated categories between the bounded derived category of coherent sheaves on the minimal resolution of \mathbb{C}^2/G and the bounded derived category of G-equivaraint sheaves on $\mathbb{C}^2$. If we enlarge to consider finite subgroups of $GL_2(\mathbb{C})$, variants of these results are known when G contains no reflections. In this talk, I will discuss recent results, inspired by work of Kawamata and Potter, giving a semiorthogonal decomposition of the derived category of G-equivariant sheaves on $\mathbb{C}^2$ whose components are in bijection with the irreducible representations of G, where G belongs to a large class of complex reflection groups of rank 2. The key input is explicit computations of group actions on the G-Hilbert schemes of Ito-Nakamura. This is joint work with Anirban Bhaduri, Yael Davidov, Eleonore Faber, Katrina Honigs, Eric Overton-Walker, and Dylan Spence.
March 18: no seminar (Spring Break)
March 25: cancelled; AG Preprint Seminar will take place during this time instead
April 1: Zhijia Zhang (New York University)
Title: Equivariant birational geometry of Fano threefolds
Abstract: The notion of G-varieties was introduced by Manin when he studied rationality problems of surfaces over perfect fields in the 1960s. A G-variety is an algebraic variety carrying a generically free regular action from a group G. There are close connections, as well as drastic differences between birational geometry of G-varieties and that of varieties over non-closed fields. In this talk, I will explore these similarities and differences with an emphasis on our work about equivariant unirationality of Fano threefolds from both algebraic and geometric aspects. This is joint work with Yuri Tschinkel and Ivan Cheltsov.
April 8: William Newman (Ohio State)
Title: Chow Groups of Moduli Spaces Via Higher Chow Groups
Abstract: One can use Bloch's higher Chow groups to compute the usual Chow groups of moduli spaces. This involves first computing the necessary higher Chow groups, and then computing the connecting homomorphism of the localization exact sequence. I will explain general techniques for performing these computations and give examples for the integral Chow groups of moduli spaces of genus 1 curves.
April 15: Robert Cass (Michigan)
Title: Affine Schubert varieties are splinters
Abstract: Schubert varieties are known to be Frobenius split in characteristic p by the work of Mehta and Ramanathan. More recently, Bhatt showed that the full flag variety for GL_n is a derived splinter by entirely different methods. We show that Bhatt's methods can be extended to general Schubert varieties and all reductive groups. Our methods apply equally well to affine Schubert varieties, which are of interest in the Langlands program and whose singularities are related to those of Shimura varieties. This is joint work with João Lourenço.
April 22: Vadim Vologodsky (UChicago)
Title: Sheared prismatization
Abstract: The prismatization construction of Drinfeld, Bhatt, and Lurie can be thought of as a deformation of the category of D-modules with locally nilpotent p-curvature on a smooth variety in characteristic p. The sheared prismatization describes a certain deformation of the category of all D-modules. In this talk, I will explain the latter and indicate some applications to the Simpson correspondence in positive and mixed characteristics.
(Joint work in progress with Bhargav Bhatt, Artem Kanaev, Akhil Mathew, Mingjia Zhang)
April 29: Louis Esser (Princeton University)
Title: Group actions and irrationality in surface families
Abstract: Celebrated work of Nicaise-Shinder and Kontsevich-Tschinkel shows that (stable) rationality specializes in smooth families. In this talk, we investigate the analogous problem for the degree of irrationality, which roughly measures how far away a variety is from being rational. We prove that this degree can only decrease under specialization for families in dimension two, provided that the map computing the degree is Galois on the very general fiber. On the other hand, we provide heuristics that suggest this result should fail if we drop the Galois assumption. This work is joint with Nathan Chen.
May 6: AG Preprint Seminar will take place during this time instead