Algebraic Geometry Seminar

A classical way of producing subvarieties of the moduli space of curves is by means of Brill-Noether theory: the locus in the moduli space of curves of genus g consisting of curves with a pencil of degree k has codimension g-2k+2. While Brill-Noether divisors have been extensively studied, little is known about higher codimension Brill-Noether loci. In this talk I will focus on the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. This is a locus of codimension two. Using the method of test surfaces, I will show how to produce a closed formula for the class of its closure in the moduli space of stable curves.

Given a homogeneous degree five polynomial W in the variables X_1, . . . , X_5, we may view W as defining a quintic hypersurface in P^4, or alternatively, as defining a singularity in [C^5/Z_5], where the group action is diagonal. In the first case, one may consider the Gromov-Witten invariants of {W=0}. In the second case, there is a way to construct analogous invariants, called FJRW invariants, of the singularity. The LG/CY correspondence conjectures that these two sets of invariants should be related. In this talk I will explain this correspondence, and its relation to a much older conjecture, the Crepant Resolution Conjecture (CRC). I will sketch a proof that the CRC is equivalent to the LG/CY correspondence in certain cases. This work is joint with Prof. Y.-P. Lee.

Let X be a non-singular rational algebraic surface (over C) equipped with a birational morphism to CP^2. Let L be a line on CP^2, and E' be the union of the strict transform of L with all but one irreducible component of the exceptional divisor. Assume that the matrix of intersection numbers of components of E' is negative definite, so that E' can be contracted to a normal analytic surface X'. Question: When is X' algebraic? I will present an answer in a geometric and then an algebraic form, and sketch the idea of the proof. The geometric answer in particular gives a correspondence between normal algebraic compactifications of C^2 with one (irreducible) curve at infinity and curves in C^2 with one (irreducible) branch at infinity. The reference for this talk is arXiv:1301.0126. If time permits, I will talk about a somewhat unexpected application in real algebraic geometry.

For a pair (X, S+B), in characteristic 0 we know that X is plt near S if and only if (S^n, B_S^n) is kit, where (K_X+S+B)|_S^n=K_S^n+B_S^n is defined by adjunction. In this talk we will show a characteristic p>0 analog of this statement.

The LG/CY correspondence was conjectured by physicists nearly twenty years ago, but has not received much attention until recently when Chiodo-Ruan proved the LG/CY correspondence for the quintic threefold. I will describe recent work joint with Mark Shoemaker in which we prove a version of the LG/CY correspondence for the mirror quintic.

The moduli space of semistable one-dimensional plane sheaves N(r,\chi) is known to be an irreducible normal projective variety of dimension r^2+1 which is also locally factorial and a Mori-Dream space of Picard number p=2. It can be shown that N(r,\chi) is a moduli space of Bridgeland semistable objects for certain stability condition and that perturbing the stability condition corresponds to run a directed MMP on N(r,\chi). By using wall-crossing techniques we can describe the exceptional loci for "most" of the flips and the divisorial contraction as well as the final model.

Using limit linear series for higher rank, I will show that given a generic curve C and under some conditions on the degree and genus, there exist a component of the expected dimension in the BN locus of stable vector bundles of rank r, degree d and k sections such that for the generic vector bundle of such component the Petri map is injective.

We prove weak approximation for a family of rationally connected varieties over a complex curve. Solution to a question by Prof Jason Starr which obstructs general weak approximation is involved.

Hurwitz numbers count degree d branched covers of the Riemann sphere by a genus g Riemann surface with prescribed ramification over one branch point and simple ramification over the others. They are intimately related to the geometry of the moduli space of curves through the famous ELSV formula. Double Hurwitz numbers similarly count covers with prescribed ramification over two points. In this talk I'm going to explain how we can describe double Hurwitz numbers as intersection numbers on the moduli space of curves using the geometry of the moduli space of relative stable maps. This helps explains geometrically the chamber/wall-crossing piecewise polynomial structure of double Hurwitz numbers. This is joint work with Renzo Cavalieri.

In tropical geometry, one often studies nodal curves by considering combinatorial invariants of their dual graph. For instance, a line bundle on the curve gives rise to a divisor on the dual graph, which, in analogy with divisors on curves, has an invariant called the rank. \\ A conjecture by Lucia Caporaso suggests that the combinatorial rank of a divisor on a graph can be described in terms of the number of global sections of line bundles on all the nodal curves which are dual to the graph. In my talk, I will discuss a number of known results and the current state of the conjecture.\\ This is joint work with Lucia Caporaso and Margarida Melo.

When considering transformation groups of rational surfaces, the most obvious cases being the group Aut(C^2) of polynomial automorphisms of the plane, or the Cremona group Bir(P^2) of birational selfmaps of the projective plane, an ubiquitous property seems to be the existence of spaces with non positive curvature on which these groups act, leading to results such as the Tits alternative or the non-simplicity. I will present an ongoing project with C. Bisi and J.-P. Furter where we obtain similar results in higher dimension: precisely we construct a CAT(0) hyperbolic square complex on which the tame group of a 3-dimensional affine quadric acts, and we deduce from this construction various properties of the group (linearizability of finite subgroups, Tits alternative, non-existence of free abelian subgroups where all non-trivial elements have dynamical degree > 1...). We also propose a similar construction for the case of Tame(C^3), whose properties are still to be explored...

TBA

The crepant transformation conjecture (CTC) predicts that K-equivalent smooth projective varieties have isomorphic quantum rings. We are interested in checking the conjecture for the local models of ordinary flops. The local model of an ordinary flop is determined by two vector bundles of the same rank over a smooth projective base. I will talk about how to reduce CTC for an arbitrary local model to local models determined by split vector bundles. This is joint work with Y.-P. Lee, H.-W. Lin and C.-L. Wang.

(joint work with Peter Scholze) The proetale topology is a Grothendieck topology that is closely related to the etale topology, yet better suited for certain "infinite" constructions, typically encountered in l-adic cohomology. I will first explain the basic definitions, with ample motivation, and then discuss applications. In particular, we will see why locally constant sheaves in this topology yield a fundamental group that is rich enough to detect all l-adic local systems through its representation theory (which fails for the groups constructed in SGA on the simplest non-normal varieties, such as nodal curves).

TBA

In the first part of the talk I will talk about the geometry of moduli spaces of pure dimension one sheaves on an Enriques or a bielliptic surface X. These moduli spaces are the relative compactified Jacobians of a linear system on X and it turns out that, when smooth, there are Calabi-Yau manifolds. In the second part of the talk, I will present another geometric construction (joint work with E. Arbarello and A. Ferretti) associated to pure dimension sheaves on X, which is a relative Prym variety whose smooth locus admits a holomorphic symplectic form. If X is an Enriques surface, I will discuss when these singular symplectic varieties admit a symplectic resolution and show that, when they do, they are deformation equivalent to Hilb^n(K3).