# Seminars & Events for Algebraic Geometry Seminar

##### Recent advances in connecting and contrasting test ideals and multiplier ideals

This talk will focus on two distinct measures of singularities: test ideals (in positive characteristic) and multiplier ideals (characteristic zero). Though known for over a decade to be related via reduction to characteristic p > 0, recent advances have provided a uniform description of these invariants using regular alterations. This description, which shall be presented in detail, simultaneously sheds new light on both the connection and differences between test and multiplier ideals. Parts of the talk are based on joint works with Manuel Blickle, Karl Schwede, and Wenliang Zhang.

##### Moduli spaces of higher dimensional varieties

The moduli spaces of smooth curves of genus at least two and its compactification, the space of stable curves, are one of the most investigated objects of algebraic geometry. In the past two decades, natural higher dimensional generalizations, the moduli space of canonically polarized manifolds and of stable schemes have been constructed. After giving a short introduction to the above spaces, I will talk about their global geometry, in particular about the hyperbolicity properties of the moduli space of canonically polarized manifolds.

##### Toric mirror maps revisited

For a compact semi-Fano toric manifold $X$, Givental's mirror theorem says that a generating function of $1$-point genus $0$ descendant Gromov-Witten invariants, the J-function of $X$, coincides up to a mirror map with a function $I_X$ which is written using the combinatorics of $X$. The procedure of obtaining the mirror map, which involves expanding $I_X$ as a suitable power series, is somewhat mysterious. In this talk we'll describe some attempts at understanding the mirror maps more geometrically.

##### Secant varieties of Segre-Veronese varieties

Secant varieties of Segre and Veronese varieties are classical objects that go back to the Italian school in the nineteen century. Surprisingly, very little is known about their equations. Inspired by experiments related to algebraic statistics, Garcia, Stillman and Sturmfels gave a conjectural description of the generators of the ideal of the secant line variety $Sec(X)$ of a Segre variety $X$. This generalizes the familiar result which states that matrices of rank two are defined by the vanishing of their $3\times 3$ minors. For a Veronese variety $X$, it was known by work of Kanev that the ideal of $Sec(X)$ is generated in degree three by minors of catalecticant matrices.

##### The Nash conjecture for surfaces

The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollár gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces.

##### BPS states, Donaldson-Thomas invariants and the Hitchin system

This talk will report on joint work with Wu-yen Chuang and Guang Pan relating the cohomology of the Hitchin system to Donaldson-Thomas theory and BPS inavariants of Calabi-Yau threefolds. A string-theoretic construction will be presented which relates refined curve-counting invariants to the work of Hausel and Rodriguez-Villegas on character varieties.

##### The Weil conjecture for singular curves

For a smooth curve, the Hilbert schemes of points are just symmetric powers of the curve, and their cohomology is easily computed in terms of H1 of the curve. In joint work with Davesh Maulik, we generalize this formula to curves with planar singularities (as conjectured by Migliorini and proved independently by Migliorini and Shende). In the singular case, the compactified Jacobian will play an important role in the formula, and our proof uses Ngô's technique from his proof of the fundamental lemma.

##### Partial desingularization of pairs

Partial desingularization consists in removing all singularities, except for those of certain class $S$, with a proper birational map that is an isomorphism over the points already in $S$. For example, if $S$ consists only of the smooth singularities, then a partial desingularization in this sense corresponds to the usual (strong) resolution of singularities. For other classes of singularities this problem has also been studied, solved or proved impossible, e.g. simple normal crossings, normal crossings, normal singularities, rational singularities, etc. It was asked by János Kollár the existence of a partial desingularization preserving the semi-simple normal crossings singularities of a pair. These are the analogous of simple normal crossings singularities in a non-normal ambient space.

##### Partial desingularization of pairs

Partial desingularization consists in removing all singularities, except for those of certain class S, with a proper birational map that is an isomorphism over the points already in S. For example, if S consists only of the smooth singularities, then a partial desingularization in this sense corresponds to the usual (strong) resolution of singularities. For other classes of singularities this problem has also been studied, solved or proved impossible, e.g. simple normal crossings, normal crossings, normal singularities, rational singularities, etc. It was asked by János Kollár the existence of a partial desingularization preserving the semi-simple normal crossings singularities of a pair. These are the analogous of simple normal crossings singularities in a non-normal ambient space.

##### Two gifts from complexity theory: $P$ v. $NP$ and matrix multiplication

I will discuss how the Geometric Complexity Theory of Mulmuley-Sohoni and the problem of determining the complexity of matrix multiplication lead to beautiful questions in algebraic geometry and representation theory.

##### Varieties fibered by good minimal models

Let $f:X->Y$ be an algebraic fiber space such that the general fiber has a good minimal model. We show that if $f$ is the Iitaka fibration then $X$ has a good minimal model. The result reduces the minimal model conjecture to the case of varieties of Kodaira dimension zero and the non-vanishing conjecture.

##### Theta Constant Identities on $Z_n$ Curves

In this talk we shall expose a concrete relation between the algebraic and transcendental parameters of a nonsingular $z_n$ curve.

A nonsingular $z_n$ curve is a compact Riemann surface with algebraic equation

$$ w^n=\prod^{j=nr−2}_{j=0}(z − \lambda_j)$$

with $\lambda_i \neq\lambda_j$ for $i \neq j$, and $r\ge 2$. Thus the Riemann surface is represented as an $n$ sheeted cover of the sphere branched over the $nr-1$ points $\lambda_0, ..., \lambda_{nr−2}$ and the point at $\infty$. We shall also here assume that $\lambda_0 = 0$, $\lambda_1 = 1$. We shall express the quantities $\lambda_i$ in terms of theta constants with rational characteristics of order $n$ and these will give rise to theta constant identities.

##### The Hodge theorem as a derived self-intersection

The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem. An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p.

##### The Stacks Project

The stacks project is a long term open source, collaborative project documenting and developing theory on algebraic stacks. I will spend a bit of time talking about what it is, who it is for, what its goals are and how it is supposed to work. More information can be found here The Stacks Project

##### Geometrically Characterizing Representation Type of Finite-dimensional Algebras

Given a finite-dimensional algebra $A$, the set of $A$-modules of a fixed dimension d can be viewed as a variety. This variety carries a group action whose orbits correspond to isomorphism classes of $A$-modules. A natural problem is to characterize various properties of an algebra $A$ in terms of its module varieties.

##### Braid Group Techniques for fundamental groups of surfaces, The K3 example

##### Tensor Products On Triangulated and Abelian Categories

##### A Frobenius Variant of Seshadri Constants

I will define a new variant of the Seshadri constant for ample line bundles in positive characteristic. We will then explore how lower bounds for this constant imply the global generation and/or very ampleness of the corresponding adjoint line bundle. As a consequence, we will deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic (even though we may lack the usual vanishing theorems). This is joint work with Mircea Mustata.

##### The Integral Hodge Conjecture For 3-Folds

The Hodge conjecture predicts which rational homology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. In other words, it is about the difference between topology and algebraic geometry. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is false in general. We discuss negative results and some new positive results on the integral Hodge conjecture for 3-folds.

##### Elliptic Functions and Equations of Modular Curves

I will talk about an old paper joint with Paul Gunnells and Sorin Popescu that explicitly describes modular curves for congruence subgroups $\Gamma_1(p)$ of $SL_2(Z)$ as intersections of quadrics in a projective space. I will aim to keep the talk accessible to graduate students.