## stanford algebraic geometry seminar 2018-19

Fridays 4-5 pm in 383-N (with exceptions)

 October 26 June Huh (Institute for Advanced Study) Distinguished Lecture 2 of 2 (TBA) November 2 Sam Payne (UT Austin and MSRI) and David Jensen (Kentucky) double-header November 30 Jack Hall (Arizona)
Posted in Uncategorized

## abstracts for 2017-18 seminars

(The seminar webpage is here.)

Sam Grushevsky (Stony Brook University)

September 29, 2017

Geometry of compactified moduli of cubic threefolds

Starting from considering the GIT compactification of the moduli of cubic threefolds, the “wonderful” compactification, which is smooth with normal crossing boundary, is constructed by an explicit sequence of blowups. We show that there exists a family of intermediate jacobians over the wonderful compactification. We compute the cohomology of the wonderful compactification by comparing it to the symplectic resolution. Based on joint works with Casalaina-Martin, Hulek, Laza

Felix Janda (University of Michigan)

October 6, 2017

Genus two curves on quintic threefolds

Virtual (Gromov-Witten) counts of maps from algebraic curves to quintic 3-folds in projective space have been of significant interest for mathematicians and physicists since the early 90s. While there are (very inefficient) algorithms for computing any specific Gromov-Witten invariant, explicit formulae are only known in genus zero and one. On the other hand, physicists have explicit conjectural formulas up to genus 51.

I will discuss a new approach to the Gromov-Witten theory of the quintic (using logarithmic geometry) which yields an explicit formula in genus two that agrees with the physicists’ conjecture.

This is based on joint works in progress with Q. Chen, S. Guo and Y. Ruan.

Remy van Dobben de Bruyn (Columbia)

October 20, 2017

Dominating varieties by liftable ones

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.

Jason Lo (Cal State Northridge)

October 27, 2017

The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations

Slope stability is a type of stability for coherent sheaves on smooth projective varieties. On a variety where the derived category of coherent sheaves admits a non-trivial autoequivalence, it is natural to ask how slope stability transforms’ to a different stability under the autoequivalence. This question also has implications for understanding the symmetries within various counting invariants. In this talk, we will give an answer to the above question for elliptic surfaces and threefolds under a Fourier-Mukai transform.

Daniel Litt (Columbia University)

December 1, 2017

Galois actions on fundamental groups

Let X be a variety over a field k, and let x be a k-rational point of X. Then the absolute Galois group of k acts on the etale fundamental group of X. If k is an arithmetically interesting field (i.e. a number field, a p-adic field, or a finite field), then this action reveals a great deal about the geometry of X; if X is a variety with an interesting fundamental group, this action reveals a great deal about the arithmetic of k.

This talk will discuss (1) joint work with Alexander Betts about the structure of Galois actions on fundamental groups, (2) how to describe invariants of these actions in terms of more geometric invariants of X, and (3) applications of this work to classical algebraic geometry, and, if time permits, arithmetic.

Pablo Solis (Stanford)

January 19, 2018

Hunting Vector Bundles on $\mathbf{P}^1 \times \mathbf{P}^1$

Motivated by Boij-Soderberg theory, Eisenbud and Schreyer conjectured there should be vector bundles on $\mathbf{P}^1 \times \mathbf{P}^1$ with natural cohomology and prescribed Euler characteristic. I’ll give some background on Boij-Soderberg theory, explain what natural cohomology means and prove the conjecture in “most” cases.

Izzet Coskun (University of Illinois at Chicago)

January 26, 2018

The geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent results concerning the Brill-Noether Theory of higher rank bundles on rational surfaces and stable cohomology of moduli spaces of sheaves. In joint work with Jack Huizenga, we characterize when the cohomology of a general stable sheaf on a Hirzebruch surface is determined by its Euler characteristic. We use these results to classify moduli spaces where the general bundle is globally generated. If time permits, I will discuss joint work with Matthew Woolf on the stable cohomology of moduli spaces on rational surfaces.

Katrina Honigs (Utah)

February 2, 2018

Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic

There are many results characterizing when derived categories of two complex surfaces are equivalent, including theorems of Bridgeland and Maciocia showing that derived equivalent Enriques or bielliptic surfaces must be isomorphic. The proofs of these theorems strongly use Torelli theorems and lattice-theoretic methods which are not available in positive characteristic. In this talk I will discuss how to prove these results over algebraically closed fields of positive characteristic (excluding some low characteristic cases). This work is joint with M. Lieblich and S. Tirabassi.

Junliang Shen (ETH)

February 9, 2018

K3 categories, cubic 4-folds, and the Beauville-Voisin conjecture

We discuss recent progress on the connection between 0-cycles of holomorphic symplectic varieties and structures of K3 categories. We propose that there exists a sheaf/cycle correspondence for any K3 category, which controls the geometry of algebraically coisotropic subvarieties of certain holomorphic symplectic varieties. Two concrete cases will be illustrated in details:
(1) the derived category of a K3 surface (joint with Qizheng Yin and Xiaolei Zhao),
(2) Kuznetsov category of a cubic 4-fold (joint with Qizheng Yin).
If time permits, we will also discuss the connection to rational curves in cubic 4-folds.

Michael Viscardi (Berkeley)

February 16, 2018

Quantum cohomology and 3D mirror symmetry

Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a semisimple group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These symplectic resolutions are expected to be symplectic duals of Nakajima quiver varieties, so that our result is an analogue of (part of) the work of Maulik and Okounkov in the symplectic dual setting.

Roya Beheshti (Washington U. St. Louis)

February 23, 2018

Moduli spaces of rational curves on hypersurfaces

I will talk about the geometry of moduli spaces of rational curves on Fano hypersurfaces and discuss some results concerning their dimension and birational geometry.

Yongbin Ruan (Michigan)

March 9, 2018

The structure of higher genus Gromov-Witten invariants of quintic 3-fold

The computation of higher genus Gromov-Witten invariants of quintic 3–fold (or compact Calabi-Yau manifold in general) has been a focal point of research of geometric and physics for more than twenty years. A series of deep conjectures have been proposed via mirror symmetry for the specific solutions as well as structures of its generating functions. Building on our initial success for a proof of genus two conjecture formula of BCOV, we present a proof of two conjectures regarding the structure of the theory. The first one is Yamaguchi-Yau’s conjecture that its generating function is a polynomial of five generators and the other one is the famous holomorphic anomaly equation which governs the dependence on four out of five generators. This is a joint work with Shuai Guo and Felix Janda.

Andrei Calderaru (Wisconsin)

April 13, 2018

Computing a categorical Gromov-Witten invariant

In his 2005 paper “The Gromov-Witten potential associated to a TCFT” Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.

In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.

Daniel Halpern-Leistner (Cornell)

April 20, 2018

What is wall-crossing?

I will discuss recent progress in understanding geometric invariant theory from an “intrinsic” perspective. This leads to a conceptually clean meta-principle for how to study the birational geometry of moduli spaces as well as a universal wall-crossing formula for the integrals of tautological K-theory classes on these moduli spaces. I will apply this perspective to the example of Bridgeland semistable complexes on an algebraic surface. The result is a relatively straightforward construction of K-theoretic Donaldson invariants, along with wall-crossing formulas for these invariants which are new-ish (conjecturally equivalent to Mochizuki’s cohomological wall crossing formulas in the context where both are defined).

Sean Howe (Stanford)

April 27, 2018

Motivic random variables and random matrices

As first shown by Katz-Sarnak, the zero spacing of L-functions of smooth plane curves over finite fields approximate the infinite random matrix statistics observed experimentally for the zero spacing of the Riemann-Zeta function (arbitrarily well by first taking the size of the finite field to infinity, and then the degree of the curve to infinity). The key geometric inputs are a computation of the image of the monodromy representation and Deligne’s purity theorem, which ensures that only the zeroth cohomology group of irreducible local systems will contribute asymptotically to the statistics. In this talk, we explain how higher order terms (i.e. the lower weight part of cohomology) can be computed starting from a simple heuristic for the number of points on a random smooth plane curve.

Isabel Vogt (MIT)

May 4, 2018, 3-4 pm

Interpolation problems for curves in projective space

In this talk we will discuss the following question: When does there exist a curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$?

Eric Larson (MIT)

May 4, 2018, 4:30-5:30 pm

The Maximal Rank Conjecture

We find the Hilbert function of a general curve of genus $g$, embedded in $\mathbb{P}^r$ via a general linear series of degree $d$. Note that Isabel Vogt’s talk earlier this afternoon is a pre-requisite for this talk.

Dori Bejleri (Brown)

May 18, 2018, 3-4 pm

Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations

A degree one del Pezzo surface is the blowup of $\mathbb{P}^2$ at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we can construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs $(X,D)$ is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of a space of weighted stable elliptic surface pairs and explain how it relates to some previous compactifications of the space of degree one del Pezzo surfaces. This is joint work with Kenny Ascher.

Francois Greer (Stony Brook)

May 18, 2018, 4:30-5:30 pm

Elliptic fibrations in the presence of singularities

The Gromov-Witten generating series for an elliptic fibration is expected to have modular properties by mirror symmetry. When the homology class in the base is irreducible and the total space is smooth, we obtain a classical modular form for the full modular group. If the base class is reducible, we expect the series to be quasi-modular. If the fibration does not admit a section, then the modular form has higher level. Both of these relaxations are related to the presence of singularities in the geometry.

Soren Galatius (Stanford)

May 25, 2018, 3-4 pm, Room TBA

$M_g$, $M_g^{trop}$, GRT, and Kontsevich’s complex of graphs

I will report on recent joint work with Melody Chan and Sam Payne on the cohomology of $M_g$ in degree $4g-6$. It is known that the rational cohomology vanishes above this degree. We prove that the rational cohomology in this degree is non-trivial for all $g \geq 7$ and that its dimension grows faster than $1.324^g +$ constant, making it asymptotically larger than the entire tautological ring and disproving a recent conjecture of Church-Farb-Putman and an older conjecture of Kontsevich. Our proof relates the weight filtration on compactly supported cohomology of $M_g$ with the moduli space of tropical curves and with the cohomology of Kontsevich’s graph complex. We then use a theorem of Willwacher to construct an injection of the Grothendieck-Teichmüller Lie algebra into $H^{2g}_c(M_g)$.

Tathagata Basak (Iowa State University)

May 25, 2018, 4:30-5:30 pm

A complex ball quotient and the monster

We shall talk about an arithmetic lattice M in $PU(13,1)$ acting on the the unit ball B in thirteen dimensional complex vector space. Let X be the space obtained by removing the hypersurfaces in B that have nontrivial stabilizer in M and then quotienting the rest by M. The fundamental group G of the ball quotient X is a complex hyperbolic analog of the braid group. We shall state a conjecture that relates this fundamental group G and the monster simple group and describe our results (joint with D. Allcock) towards this conjecture.

The discrete group M is related to the Leech lattice and has generators and relations analogous to Weyl groups. Time permitting, we shall give a second example in $PU(9,1)$ related to the Barnes-Wall lattice for which there is a similar story.

Posted in seminars

## stanford algebraic geometry seminar 2017-18

Fridays 4-5 pm in 383-N (with exceptions)

 September 29 Samuel Grushevsky (Stony Brook University) Geometry of compactified moduli of cubic threefolds October 6 Felix Janda (University of Michigan) Genus two curves on quintic threefolds October 13 no seminar (WAGS weekend) Weekend of October 14-15 Western Algebraic Geometry Symposium, at UCLA October 20 Remy van Dobben de Bruyn (Columbia University) Dominating varieties by liftable ones October 27 Jason Lo (Cal State Northridge) The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations November 10 November 17 (probably no seminar, Ravi away) November 24 no seminar (Thanksgiving break) December 1 Daniel Litt (Columbia University) Galois actions on fundamental groups January 19 Pablo Solis (Stanford) Hunting Vector Bundles on $\mathbf{P}^1 \times \mathbf{P}^1$ January 26 Izzet Coskun (UIC) The geometry of moduli spaces of sheaves on surfaces February 2 Katrina Honigs (Utah) Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic February 9 Junliang Shen (ETH) K3 categories, cubic 4-folds, and the Beauville-Voisin conjecture February 16 Michael Viscardi (Berkeley) Quantum cohomology and 3D mirror symmetry February 23 Roya Beheshti (Washington U. St. Louis) Moduli spaces of rational curves on hypersurfaces March 9 Yongbin Ruan (Michigan) The structure of higher genus Gromov-Witten invariants of quintic 3-fold April 13 Andrei Caldararu (Wisconsin) Computing a categorical Gromov-Witten invariant April 20 Daniel Halpern-Leistner (Cornell) What is wall-crossing? April 27 Sean Howe Motivic random variables and random matrices May 4 (3-4 pm, in 380-W) Isabel Vogt (MIT) Interpolation problems for curves in projective space May 4 (4:30-5:30 pm, in 383-N) Eric Larson (MIT) The Maximal Rank Conjecture May 11 Sheldon Katz (UIUC) (joint with physics) May 18 (3-4 pm) Dori Bejleri (Brown) Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations May 18 (4:30-5:30 pm) Francois Greer (Stony Brook) Elliptic fibrations in the presence of singularities May 25 (3-4 pm, location TBA) Soren Galatius (Stanford) $M_g$, $M_g^{trop}$, the Grothendieck-Teichmuller group, and Kontsevich’s complex of graphs May 25 (4:30-5:30 pm) Tathagata Basak (Iowa State) A complex ball quotient and the monster
Posted in seminars

## abstracts for 2016-17 seminars

(The seminar webpage is here.)

Michael Kemeny (Stanford)
October 7, 2016

Syzygies, scrolls and Hurwitz spaces

A famous conjecture of Mark Green predicts a close relationship between the geometry of a curve and the algebraic properties of its coordinate ring. Namely, the Clifford index of the curve should equal the length of the linear part of the resolution of its coordinate ring under the canonical embedding. A conjecture of Schreyer goes beyond this by specifying that the last piece of the linear part should moreover tell you whether or not the curve has a unique pencil of minimal degree. We will discuss a proof of Schreyer’s conjecture for general curves of prescribed gonality, obtained jointly with Gavril Farkas. Two of the key actors in this story are the scroll associated to a pencil and the geometry of Hurwitz space.  (poster)

Christian Schnell (Stony Brook University)
October 21, 2016

Pushforwards of pluricanonical bundles and morphisms to complex abelian varieties

In the past few years, people working on the analytic side of algebraic geometry have obtained two important new results: a version of the Ohsawa-Takegoshi extension theorem with sharp estimates (Blocki, Guan-Zhou), and the existence of canonical singular hermitian metrics on pushforwards of relative pluricanonical bundles (Berndtsson, Paun, Takayama, and others). In this talk, I will explore some consequences of their work for the study of morphisms to complex abelian varieties, including the recent proof of Iitaka’s conjecture over abelian varieties (Cao-Paun).  (The talk will be understandable without any background in analysis.)

Wenhao Ou (UCLA)
October 28, 2016

Fano varieties where all pseudoeffective divisors are also numerically effective

We recall that a divisor in a smooth projective variety is said to be numerically effective (or nef) if it meets each curve with non negative intersection number, and is called pseudoeffective if it is the limit of effective Q-divisor classes. Both of these properties are ways in which a divisor can be in some sense “positive”. A nef divisor is always pseudoeffective, but the converse is not true in general. A Fano varity is a special variety whose anti-canonical divisor is ample. From the Cone Theorem, it turns out that the geometry of a Fano variety is closely related to its nef divisors. In this talk, we will consider Fano varieties such that all pseudoeffective divisors are nef. Wiśniewski shows that the Picard number of such a variety is at most equal to its dimension. Druel classifies these varieties when these two numbers are equal. We classify the case when the Picard number is equal to the dimension minus 1.

November 4, 2016

Isospectrality of compact locally symmetric spaces and weak commensurability of arithmetic groups

Quotients of symmetric spaces of semi-simple Lie groups by torsion-free arithmetic subgroups are particularly interesting Riemannian manifolds which can be studied by using diverse techniques coming from the theories of Lie Groups, Lie Algebras, Algebraic Groups and Automorphic Forms. In my talk, I will discuss a well-known problem which was formulated by Mark Kac as “Can one hear the shape of a drum?”, and its solution, for arithmetic quotients of symmetric spaces, obtained in a joint paper (in Publ Math IHES, vol 109) with Andrei Rapinchuk. For its solution, we introduced a notion of “weak commensurability” of arithmetic, and more general Zariski-dense, subgroups and derive very strong consequences of weak commensurability.

Donghai Pan (Stanford)
November 11, 2016

Galois cyclic covers of the projective line and pencils of Fermat hypersurfaces

Classically, there are two objects that are particularly interesting to algebraic geometers: hyperelliptic curves and quadrics. The connection between these two seemingly unrelated objects was first revealed by M. Reid, which roughly says that there’s a correspondence between hyperelliptic curves and pencil of quadrics. I’ll give a brief review of Reid’s work and then describe a higher degree generalization of the correspondence.

Giulia Sacca (Stony Brook)
December 2, 2016

Intermediate Jacobians and hyperkahler manifolds

In recent years, there have been an increasing number of connections between cubic fourfolds and hyperkahler manifolds. The first instance of this was noticed by Beauville-Donagi, who showed that the Fano varieties of lines on a cubic fourfolds X is holomorphic symplectic. The aim of the talk is to describe another instance of this phenomenon, which is carried out in joint work with Laza and Voisin. The resulting hyperkahler manifold is fibered in intermediate Jacobians and is deformation equivalent to O’Grady’s ten-dimensional example. I will also present a more recent proof of this result, which is obtained in joint work with Laza, Kollár, and Voisin.

Hannah Larson (Harvard)
January 13, 2017, 4-4:45 pm

Lines on hypersurfaces with certain normal bundles

Let $X$ be a smooth hypersurface. The Fano scheme of lines $F(X)$ is the parameter space of all lines $L \subset X$. Given such a line, the normal bundle of $L$ in $X$ controls the deformation theory of $L$ in $X$, and thus provides local information about $F(X)$ near $L$. Being a vector bundle on ${\mathbf{P}}^1$, the normal bundle of $L$ in $X$ always splits as a direct sum of line bundles. In this talk, we consider natural subschemes of $F(X)$ parameterizing lines $L$ whose normal bundle in $X$ has a certain splitting type.

Gavril Farkas (Humboldt University)
January 13, 2017, 5-6 pm

K3 surfaces of genus 14 via cubic fourfolds

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera $g<21$, with the exception of the case $g=14$. Using Hassett's identification between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. The proof relies on a degenerate version of Mukai's structure theorem for K3 surfaces of genus 8. This is joint work with Verra.

Ashvin Swaminathan (Harvard)
Thursday, January 19, 2017, 12:15-1 pm in 384-I

Inflection Points of Linear Systems on Families of Curves

It is a classic theorem in enumerative geometry that a general plane curve of degree $d$ has exactly $3d(d-2)$ flex points (these are inflection points at which the tangent line has contact of order $3$). Given this result, there are two natural generalizations to consider: (1) what can we say about inflection points of higher contact order, and (2) what happens when we look at such inflection points in families of curves acquiring a singularity? In this talk, I will discuss joint work with Anand Patel, in which we develop a method for answering these more general questions. Moreover, I will describe how to apply our method to tackle three interesting problems: (1) counting hyperflexes in a general pencil of plane curves, (2) describing the analytic-local behavior of the divisor of flexes in a family of plane curves acquiring a nodal singularity, and (3) computing the divisors of Weierstrass points of arbitrary order on the moduli space of curves.

Jake Levinson (Michigan)
January 20, 2017, 3:45-4:45 pm

Boij-Söderberg Theory for Grassmannians

Boij-Söderberg theory is a structure theory for syzygies of graded modules: a near-classification of the possible Betti tables of such modules (these tables record the degrees of generators in a minimal free resolution). One of the surprises of the theory was the discovery of a “dual” classification of sheaf cohomology tables on projective space.

I’ll tell part of this story, then describe some recent extensions of it to the setting of Grassmannians. Here, the algebraic side concerns modules over a polynomial ring in $kn$ variables, thought of as the entries of a $k \times n$ matrix. The goal is to classify “$GL(k)$-equivariant Betti tables”, recording the syzygies of equivariant modules, and to relate them to sheaf cohomology tables on the Grassmannian $Gr(k,n)$. This work is joint with Nic Ford and Steven Sam.

Ben Bakker (Georgia)
January 20, 2017, 5-6 pm

A global Torelli theorem for singular symplectic varieties

Holomorphic symplectic manifolds are the higher-dimensional analogs of K3 surfaces and their local and global deformation theories enjoy many of the same nice properties.  By work of Namikawa, some aspects of the story generalize to singular symplectic varieties, but the lack of a well-defined period map means the moduli theory is ill-defined.  In joint work with C. Lehn, we consider locally trivial deformations—deformations along which the singularities don’t change — and show that in this context most of the results from the smooth case extend.  In particular, we prove a version of the global Torelli theorem and derive some applications to the geometry of birational contractions of moduli spaces of vector bundles on K3 surfaces.

Srikanth Iyengar (Utah)
January 27, 2017

A local Serre duality for modular representations of finite groups (and group schemes)

This talk will be about the representations of a finite group (or a finite group scheme) G defined over a field k of positive characteristic. In recent work, Dave Benson, Henning Krause, and Julia Pevtsova, and I discovered that stable module category of finite dimensional representations of G has local Serre duality. My plan is to explain what this means and also present some of the ideas, mostly from commutative algebra, that go into its proof.

Steven Sam (Wisconsin)
February 3, 2017

Secant varieties of Veronese embeddings

Given a projective variety X over a field of characteristic 0, and a positive integer r, we study the rth secant variety of Veronese re-embeddings of X. In particular, I’ll explain recent work which shows that the degrees of the minimal equations (and more generally, syzygies) defining these secant varieties can be bounded in terms of X and r independent of the Veronese embedding. This is based on http://arxiv.org/abs/1510.04904 and http://arxiv.org/abs/1608.01722.

February 10, 2017

Double Ramification Cycles and Tautological Relations

Tautological relations are certain equations in the Chow ring of the moduli space of curves. I will discuss a family of such relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line. These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations. This is joint work with various subsets of S. Grushevsky, F. Janda, X. Wang, and D. Zakharov.

Ionut Ciocan-Fontanine (Minnesota)
February 17, 2017

Wall-crossing in quasimap theory

Quasimap theory is concerned with curve counting on certain GIT quotients. In fact, one has a family of curve counting theories, with Gromov-Witten included, depending on the linearization in the GIT problem. I will present a wall-crossing formula, in all genera and at the level of virtual classes, as the size of the linearization changes. Some numerical consequences will be discussed as well. The talk will focus primarily on the recently established case of complete intersections in projective space (in this case, stable quasimaps coincide with stable quotients). This is joint work with Bumsig Kim.

Dusty Ross (SFSU)
February 24, 2017

Genus-One Landau-Ginzburg/Calabi-Yau Correspondence

First suggested by physicists in the late 1980’s, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence, which is determined from genus-zero data alone. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.

Dhruv Ranganathan (MIT)
April 14, 2017

A Brill-Noether theorem for curves of a fixed gonality

The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill–Noether theorem, which determines the dimensions of the Brill–Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus”. The proof blends recent advances in tropical linear series theory, Berkovich geometry, and ideas from logarithmic Gromov-Witten theory. This is joint work with Dave Jensen.

Erik Carlsson (UC Davis)
April 21, 2017

Geometry behind the shuffle conjecture

The original “shuffle conjecture” of Haglund, Haiman, Loehr, Ulyanov, and Remmel predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A, in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the ideas that went into my recent proof of this conjecture with Anton Mellit, namely the construction of a new algebra which has many elements in common with DAHA’s, and which has been expected to have a geometric construction. I will then describe a current project with Eugene Gorsky and Mellit, in which we have discovered the desired action of this algebra on the torus-equivariant K-theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals of finite codimension in C[x,y].

Arnav Tripathy (Harvard)
May 12, 2017

Motivic Donaldson-Thomas invariants of K3 times an elliptic curve

I’ll describe a new chapter in the enumerative geometry of the K3 surface and its product with an elliptic curve in a long line of extensions starting from the classic Yau-Zaslow formula for counts of rational nodal curves. In particular, I’ll describe a string-theoretic prediction for the threefold’s motivic Donaldson-Thomas invariants given the Hodge-elliptic genus of the K3, a new quantity interpolating between the Hodge polynomial and the elliptic genus.

John Lesieutre (UIC)
May 19, 2017

A projective variety with discrete, non-finitely generated automorphism group

I will outline the construction of a projective variety over $\mathbf{C}$ for which the group of automorphisms is discrete, but not finitely generated. I’ll also discuss a related example of a variety over $\mathbf{R}$ with infinitely many non-isomorphic $\mathbf{C}/\mathbf{R}$-forms.

Yuchen Liu (Princeton)
May 26, 2017

Construction of hyperbolic cyclic covers

A complex algebraic variety is Brody hyperbolic if there are no non-constant holomorphic maps from the complex plane to the variety. As shown by Duval, Shiffman, Zaidenberg and many other people, degeneration methods are very useful in constructing Brody hyperbolic varieties. Using degeneration to the normal cone, we construct lots of examples of Brody hyperbolic cyclic covers when their branch loci have sufficiently large degree. If time permits, I will also discuss a different approach to construct hyperbolic double covers of $\mathbf{P}^2$ and Hirzebruch surfaces with the smallest possible degree of branch loci.

Posted in seminars

## stanford algebraic geometry seminar 2016-17

Fridays 4-5 pm in 383-N (with exceptions)

 October 7 Michael Kemeny Syzygies, scrolls and Hurwitz spaces October 14 no seminar (WAGS weekend) October 15-16 weekend Enrico Arbarello, Sapienza Universita di Roma/Stony Brook, Emily Clader, San Francisco State University, Luis Garcia, University of Toronto, Diane Maclagan University of Warwick, Sandra Di Rocco, KTH, and Brooke Ullery, University of Utah Western Algebraic Geometry Symposium (at the Colorado State University) October 21 Christian Schnell (Stony Brook University) Pushforwards of pluricanonical bundles and morphisms to complex abelian varieties October 28 Wenhao Ou (UCLA) Fano varieties where all pseudoeffective divisors are also numerically effective November 4 Gopal Prasad (University of Michigan) Isospectrality of compact locally symmetric spaces and weak commensurability of arithmetic groups November 11 Donghai Pan (Stanford) Galois cyclic covers of the projective line and pencils of Fermat hypersurfaces November 25 no seminar (Thanksgiving break) December 2 Giulia Sacca (Stonybrook University) Intermediate Jacobians and hyperKahler manifolds January 13, 4-4:45 pm Hannah Larson (Harvard) Lines on hypersurfaces with certain normal bundles January 13, 5-6 pm Gavril Farkas (Humboldt University) K3 surfaces of genus 14 via cubic fourfolds Thursday, January 19, 12:15-1 pm in 384-I Ashvin Swaminathan (Harvard) Inflection Points of Linear Systems on Families of Curves January 20, 3:45-4:45 pm Jake Levinson (Michigan) Boij-Soderberg Theory for Grassmannians January 20, 5-6 pm Ben Bakker (Georgia) A global Torelli theorem for singular symplectic varieties January 27 Srikanth Iyengar (Utah) A local Serre duality for modular representations of finite groups (and group schemes) February 3 Steven Sam (Wisconsin) Secant varieties of Veronese embeddings February 10 Emily Clader (SFSU) Double ramification cycles and tautological relations February 17 Ionut Ciocan-Fontanine (Minnesota) Wall-crossing in quasimap theory February 24 Dusty Ross (SFSU) Genus-One Landau-Ginzburg/Calabi-Yau Correspondence April 7 no seminar April 8-9 Western Algebraic Geometry Symposium (at UBC) April 14 Dhruv Ranganathan (MIT) A Brill-Noether theorem for curves of a fixed gonality April 21 Erik Carlsson (Davis) Geometry behind the shuffle conjecture May 12 Arnav Tripathy (Harvard) Motivic Donaldson-Thomas invariants of K3 times an elliptic curve May 19 John Lesieutre (UIC) A projective variety with discrete, non-finitely generated automorphism group May 26 Yuchen Liu (Princeton) Construction of hyperbolic cyclic covers
Posted in seminars

## abstracts for 2015-16 seminars

(The seminar webpage is here.)

Mihnea Popa (Northwestern University)
September 25, 2015

Positivity for Hodge modules and geometric applications

M. Saito’s theory of Hodge modules provides a powerful generalization of classical Hodge theory that is beginning to find basic applications to birational geometry. One of the main reasons for this is that generalizations of Hodge bundles and de Rham complexes arising in this context satisfy analogues of well-known vanishing and positivity theorems. I will review a number of such results that have been obtained recently, and how they can be applied to deduce new statements regarding holomorphic one-forms and families of varieties of general type. Much of this work is joint with C. Schnell. This talk is intended for a wide algebro-geometric audience, and will contain a review of necessary background from Hodge theory.

Tony Pantev (University of Pennsylvania)
October 2, 2015

Symplectic geometry, foliations, and potentials

I will explain how Lagrangian foliations in derived algebraic symplectic geometry give rise to global potentials. I will also give natural constructions of isotropic foliations on moduli
spaces and will discuss the associated potentials. I will give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.  This talk is intended for a general algebro-geometric audience, and no advanced background will be assumed.

Alex Perry (Harvard)
October 30, 2015

Categorical joins

Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of
classical projective duality. I will describe a categorical version of the classical join of two projective varieties, and its relation to homological projective duality. I will discuss some applications to the structure of the derived categories of Fano varieties and to derived equivalences of Calabi-Yau varieties. This is work in progress with Alexander Kuznetsov.

Zijun Zhou (Columbia)
November 13, 2015

Relative orbifold Donaldson-Thomas theory and local gerby curves

In this talk I will introduce the generalization of relative Donaldson-Thomas theory to 3-dimensional smooth Deligne-Mumford stacks. We adopt Jun Li’s construction of expanded pairs and degenerations and prove an orbifold DT degeneration formula. I’ll also talk about the application in the case of local gerby curves, and its relationship to the work of Okounkov-Pandharipande and Maulik-Oblomkov.

Vikraman Balaji (Chennai Mathematical Institute)
November 20, 2015, 2:30-3:30

Degeneration of moduli of Higgs bundles on curves

In this this talk I will discuss the construction of a degeneration of the moduli space of Higgs bundles on smooth curves, as the smooth curve $X$ degenerates to a nodal curve with a single node. As an application we get new compactifications of the Picard variety for smooth curves degenerating to irreducible nodal curves (with multiple nodes) which have analytic normal crossing singularities.

Jose Rodriguez (University of Chicago)
November 20, 2015

Numerically computing Galois groups with Bertini.m2

Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile.

Yukinobu Toda (Institute for the Physics and the Mathematics of the Universe, University of Tokyo)
December 4, 2015, 2:30-3:30 (because of the department party)

Non-commutative deformations and Donaldson-Thomas invariants

In this talk, I will show the existence of certain global non-commutative structures on the moduli spaces of stable sheaves on algebraic varieties, whose formal completion at a closed point gives the pro-representable hull of the non-commutative deformation functor of the sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. I will then introduce the generating series through integrations over Hilbert schemes of points on these NC structures. When the underlying variety is a Calabi-Yau 3-fold, and the moduli space of stable sheaves satisfy some assumptions, this generating series admits a product expansion described by generalized DT invariants. This formula explains the dimension formula of Donovan-Wemyss’s contractions algebras for floppable curves on 3-folds in terms of genus zero Gopakumar-Vafa invariants.

Aaron Landesman (Harvard)
Tuesday, January 5, 2016 in 384-H (Student Algebraic Geometry Seminar)

Interpolation of Projective Varieties

In this talk, we discuss interpolation of projective varieties through points. It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. More recently, it was shown that one can always find nonspecial curves through the expected number of general points. We consider the generalization of this question to varieties of all dimensions and explain why rational normal scrolls satisfy interpolation. Time permitting, we’ll also discuss joint work with Anand Patel on interpolation for del Pezzo surfaces and present several interesting open interpolation problems. We’ll place particular emphasis on explaining the standard techniques used to solve interpolation problems: deformation theory, specialization, degeneration, and the gale transform.

Tonghai Yang (Wisconsin)
January 15, 2016, 2:30-3:30 pm (joint with Number Theory)

Generating series of arithmetic divisors on a Shimura variety of unitary type $(n-1, 1)$

In this talk we will define a generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1), and give a rough idea how to prove that it is modular. This can be viewed as generalization of modular form of weight 2 of Heeger divisors used in Gross-Zagier formula and its generalization Yuan-Zhang-Zhang formula. If time permits, I will mention its application to a special case of Colmez conjecture and integral’ generalization of the Gross-Zagier and Yuan-Zhang-Zhang formula. This is a joint work with Bruinier, Howard, Kudla, and Rapoport.

Sheldon Katz (UIUC)
January 15, 2016

Stable pair invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms

I explain the conjectural description of the generating function of stable pair invariants of elliptically fibered Calabi-Yau threefolds with fixed base class and variable fiber class in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base class.  This talk is based on joint work with Albrecht Klemm and Minxin Huang.

Jacob Tsimerman (Toronto)
January 22, 2016

Bounding 2-torsion in class groups

(joint with Bhargava, Shankar, Taniguchi, Thorne, and Zhao) Zhang’s conjecture asserts that for fixed positive integers $m$, $n$, the size of the m-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a handful of cases, the best known result towards this conjecture is the ”convex” bound given by the Brauer-Siegel Theorem. We make progress on this conjecture by giving a ”subconvex” bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of $n$. The proof is surprisingly elementary, and we give several applications of this result stemming from the case of cubic fields, including improved bounds on the number of $A_4$ fields, and on the number of integer points an elliptic curve can have. Along the way, we prove a surprising result on the shape of the lattice of the ring of integers of a number field. Namely, we show that such a lattice is very limited in how ‘skew’ it can be.

Vivek Shende (Berkeley)
January 29, 2016

Cluster varieties from Legendrian knots

We give a uniform geometric explanation for the existence of cluster structures on the positroid strata, character varieties, wild character varieties, etc.

The point is that such spaces can be identified as the moduli spaces of sheaves with microsupport in an appropriate Legendrian knot. In general, an exact Lagrangian with asymptotics in the knot gives rise to an algebraic torus chart on the moduli space. We construct such fillings for alternating knots, so get charts in correspondence with alternating representatives of the knot in question. Such can be enumerated by bicolored graphs, and we recover the various aspects of the bicolored graph theory — the boundary measurement map of Postnikov; the cluster structure on the positroid varieties, the analogous construction on the character varieties due to Fock and Goncharov — by computing the Floer homology between our fillings.

This talk presents joint work with David Treumann, Harold Williams, and Eric Zaslow.

Izzet Coskun (University of Illinois at Chicago)
April 15, 2016

Birational geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent developments in the birational geometry of moduli spaces of sheaves on surfaces motivated by Bridgeland stability conditions. After reviewing joint work with Arcara, Bertram, Huizenga and Woolf in the case of the projective plane, I will describe how to use Bridgeland stability conditions to construct nef divisors on moduli spaces of sheaves on surfaces. I will illustrate the theory with examples on the projective plane, the quadric surface and some general type surfaces. This is joint work with Jack Huizenga.

Zhiwei Yun (Stanford)
April 29, 2016

Intersection numbers and higher derivatives of L-functions for function fields

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under some unramifiedness assumptions. Our formula relates the self-intersection number of certain cycles on the moduli of Drinfeld Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

Shouwu Zhang (Princeton)
Thursday, May 12, 2016, 4:30 pm, 380-W

Lecture 1 of Distinguished Lecture Series: Rational points: ABC conjecture and Szpiro’s conjecture

I will talk about ABC conjecture and its relation to Szpiro’s conjecture, Milnor’s proof of Szpiro’s inequality (before it was known as Szpiro’s conjecture) and its relation to hyperbolic geometry, and relation between Szpiro’s conjecture and Landau—Siegel conjecture on zeros of L-functions.

Shouwu Zhang (Princeton)
Friday, May 13, 2016, 2:30 pm, 383-N

Lecture 2 of Distinguished Lecture Series: Torsion points and preperiodic points: Manin—Mumford’s conjecture and its dynamical analogue

I will talk about techniques used in different proofs of Manin-Mumford’s conjecture and its analogue in dynamical system: p-adic rigid geometry (Raynaud), o-minimality geometry (Pila—Zannier), Arakelov geometry (Ullmo—Zhang), and perfectoid geometry (Xie).

Shouwu Zhang (Princeton)
Monday, May 16, 2016, 4 pm, 384-H

Lecture 1 of Distinguished Lecture Series: CM points and derivatives of L-dunctions: Andre—Oort’s conjecture and Colmez’ conjecture

I will talk about recent work of Tsimerman about reducing Andre—Oort’s conjecture to an averaged version of Colmez’ conjecture, and some related work on derivatives of L-functions by Zhiwei Yun and Wei Zhang using Drinfeld’s moduli of Shtukas, and by Xinyi Yuan using Shimura curves.

Ronen Mukamel (Rice)
May 20, 2016 (joint with Informal Geometry and Topology Seminar)

Equations for real multiplication in genus two and applications to Teichmuller curves

I will describe methods for computing with genus two curves whose Jacobians admit real multiplication and discuss applications of these methods to Teichmuller curves. This is joint work with Abhinav Kumar.

Colleen Robles (Duke)
May 27, 2016

Degenerations of Hodge structures

Two motivating questions from algebraic geometry are: how can a smooth projective variety degenerate? and what are the “relations” between two such degenerations? One way to gain insight into these questions is to ask the analogous questions of invariants associated with the smooth projective varieties. In this case, the invariant that we have in mind is a polarized Hodge structure. (And indeed detailed analysis of degenerations of polarized Hodge structures can be used to better understand degeneration of smooth projective varieties, and moduli spaces and their compactifications.)

I will explain how the work of Cattani, Kaplan and Schmid allows us to view a polarized limiting mixed Hodge structure (PLMHS) as a degeneration of a polarized Hodge structure. There is a notion of “polarized relation” between PLMHS that encodes information on how varieties may degenerate within a family. I will give a classification of PLMHS and their polarized relations in terms of Hodge diamonds (discrete data associated with a PLMHS), effectively answering the Hodge-theoretic analogs of the two motivating questions above. This is joint work with Matt Kerr.

Posted in Uncategorized

## stanford algebraic geometry seminar 2015-16

Fridays 3:45-4:45 in 383-N (with exceptions)

 September 25 at 3 pm Mihnea Popa (Northwestern) Positivity for Hodge modules and geometric applications October 2 Tony Pantev (U Penn) Symplectic geometry, foliations, and potentials October 17-18 weekend Western Algebraic Geometry Symposium (at the University of Washington) Aravind Asok (USC), Yiwei She (Columbia), Rekha Thomas (UW), Nikolaos Tziolas (University of Cyprus), Alena Pirutka (Courant), Brian Osserman (UC Davis), Valery Alexeev (U Georgia) October 23 no seminar October 30 Alex Perry (Harvard) Categorical joins November 13 Zijun Zhou (Columbia) Relative orbifold Donaldson-Thomas theory and local gerby curves November 20 (2:30-3:30) Vikraman Balaji (Chennai Mathematical Institute) Degeneration of moduli of Higgs bundles on curves November 20 (3:45-4:45) Jose Rodriguez (Chicago) Numerically computing Galois groups with Bertini.m2 November 27 no seminar (Thanksgiving break) December 4 (2:30-3:30, because of the department party) Yukinobu Toda (Institute for the Physics and Mathematics of the Universe) Non-commutative deformations and Donaldson-Thomas invariants January 5 (Student Algebraic Geometry Seminar outside speaker), in 383-H Aaron Landesman (Harvard) Interpolation of Projective Varieties January 15 (joint Algebraic Geometry – Number theory seminar),  2:30-3:30 Tonghai Yang (Wisconsin) Generating series of arithmetic divisors on a Shimura variety of unitary type (n-1, 1) January 15 Sheldon Katz (UIUC) Stable pair invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms January 22 (joint Algebraic Geometry – Number theory seminar) Jacob Tsimerman (Toronto) Bounding 2-torsion in class groups January 29 Vivek Shende (Berkeley) Cluster varieties from Legendrian knots April 15 Izzet Coskun (UIC) Birational geometry of moduli spaces of sheaves on surfaces April 29 Zhiwei Yun (Stanford) Intersection numbers and higher derivatives of L-functions for function fields Thursday May 12, 4:30 pm in 380-W Shouwu Zhang (Princeton) Rational points: ABC conjecture and Szpiro’s conjecture (first in a Distinguished Lecture Series) May 13, 2:30 pm in 383-N Shouwu Zhang (Princeton) Torsion points and preperiodic points: Manin—Mumford’s conjecture and its dynamical analogue (second in a Distinguished Lecture Series) Monday May 16, 4 pm in 384-H Shouwu Zhang (Princeton) CM points and derivatives of L-dunctions: Andre—Oort’s conjecture and Colmez’ conjecture (third in a Distinguished Lecture Series) May 20 (joint with the Informal Geometry and Topology Seminar) Ronen Mukamel (Rice) Equations for real multiplication in genus two and applications to Teichmuller curves May 27 Colleen Robles (Duke) Degenerations of Hodge structures

Fall 2016 speakers:  Christian Schnell (Stony Brook), Giulia Sacca (Stony Brook), and (although he doesn’t know it yet) Michael Kemeny (Stanford)

Posted in seminars | Tagged

## abstracts for 2014-15 seminars

(The seminar webpage is here.)

Jarod Alper (Australian National University)
September 26, 2014

Associated forms in classical invariant theory

There is an interesting map which assigns to a homogeneous form $f$ on $\mathbb{C}^n$ of degree $d$ with non-vanishing discriminant, a certain form on $\mathbb{C}^n$ of degree $n(d-2)$, which is the Macaulay inverse system of the Milnor algebra of $f$.  It was conjectured in a recent paper by M. Eastwood and A. Isaev that all absolute classical invariants of forms on $\mathbb{C}^n$ of degree $d$ can be extracted from those of forms of degree $n(d-2)$ via this map.  This surprising conjecture was motivated by the well-known Mather-Yau theorem for isolated hypersurface singularities.  I will report on joint work with A. Isaev which settles this conjecture in full generality and proves a stronger statement in the case of binary forms.

Jim Bryan (University of British Columbia)
October 24, 2014

Donaldson-Thomas theory of local elliptic surfaces via the topological vertex

Donaldson-Thomas (DT) invariants of a Calabi-Yau threefold X are fundamental quantum invariants given by (weighted) Euler characteristics of the Hilbert schemes of X. We compute these invariants for the case where X is a so-called local elliptic surface — it is the total space of the canonical line bundle over an elliptic surface. We find that the generating functions for the invariants admit a nice product structure. We introduce a new technique which allows us to use the topological vertex in this computation — a tool which previously could only be used for toric threefolds. As a by product, we discover surprising new identities for the topological vertex. This is joint work with Martijn Kool, with an assist from Ben Young.

October 31, 2014

Interpolation and vector bundles on curves

We aim to address the following: When is there a (smooth) curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$. Generalizations ask for the dimension of such curves, or replace the point incidence conditions with higher dimensional linear spaces. We will start by relating these statements to a property of the normal bundle of curves in projective space. Next, we will show how to address these questions for $r = 3$ and $d \geq g + 3$. The demonstrated techniques generalize significantly and lead to an answer to our question for $d \geq g + r$. This is joint work with E. Larson and D. Yang.

Yefeng Shen (Stanford)
November 7, 2014

WDVV equations and Ramanujan identities

The occurrence of modular forms and quasi-modular forms in Gromov-Witten theory is an interesting phenomenon. I will present the following work joint with Jie Zhou. We show that the WDVV equations for elliptic orbifolds are equivalent to the Ramanujan identities for some modular groups. We then apply this to prove the genus zero Gromov-Witten correlation functions for all elliptic orbifolds are quasi-modular forms. Combining with the tautological relations on the moduli space of pointed curves, we also obtain the modularity for all genera. This generalizes an earlier result of Milanov–Ruan and solves a modularity conjecture for the Gromov-Witten theory of the elliptic orbifold curve with four $\mathbb{Z}_2$-orbifold points.

Yiwei Shi (University of Chicago)
November 14, 2014

The Shafarevich conjecture for K3 surfaces

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces.

Sabin Cautis (University of British Columbia)
November 21, 2014

Categorical Heisenberg actions on Hilbert schemes of points

We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE surfaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and derived categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

François Charles (Paris Sud and MIT)
December 5, 2014, 1:15 pm (unusual time!)

Geometric boundedness results for K3 surfaces

Tate’s conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin’s trick. We will discuss such geometric boundedness statements for K3 surfaces over arbitrary fields and holomorphic symplectic varieties, with application to direct proofs of the Tate conjecture for K3 surfaces that do not involve the Kuga-Satake correspondence.

Dan Edidin (Missouri)
January 9, 2015

Strong regular embeddings and the geometry of hypertoric stacks

We explain how the notion of “strong regular embeddings” can be used to compare the geometry of a stack to that of a regularly embedded substack. This theory can be applied to understand the relationship between singular hypertoric varieties and singular Lawrence toric varieties. While this talk is about stacks, the motivating ideas come from simple observations about invariant rings for actions of finite groups.

Evan O’Dorney (Harvard)
January 16, 2015

Canonical rings of $\mathbf{Q}$-divisors on $\mathbf{P}^1$

The canonical ring $S_D = \bigoplus_{d\geq 0} H^0(X, \lfloor dD\rfloor)$ of a divisor $D$ on a projective curve $X$ is a natural object of study; when $D$ is a $\mathbf{Q}$-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. I will speak about my results from last summer’s Emory REU concerning the generators and relations of $S_D$ for the simplest curve $X = \mathbf{P}^1$. When $D$ contains at most two points, I give a complete description of $S_D$; for general $D$, I give bounds on the generators and relations. I have also proved that the generators (for at most five points) and a Groebner basis of relations between them (for at most four points) depend only on the coefficients in the divisor $D$, not its points or the characteristic of the ground field, and I conjecture that the minimal system of relations varies in a similar way.

Mark Shoemaker (Utah)
January 23, 2015

A proof of the LG/CY correspondence via the crepant resolution conjecture

Given a homogeneous degree five polynomial $W$ in the variables $X_1$, …, $X_5$, we may view $W$ as defining a quintic hypersurface in $\mathbf{P}^4$ or alternatively, as defining a singularity in $[ \mathbf{C}^5/\mathbf{Z}_5]$ 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 states that these two sets of invariants are 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 using a generalization of the “quantum Serre-duality” of Coates-Givental. This work is joint with Y.-P. Lee and Nathan Priddis.

January 30, 2015, 3:20-4:20

Moduli of degree 4 K3 surfaces revisited

For low degree K3 surfaces there are several way of constructing and compactifying the moduli space (via period maps, via GIT, or via KSBA). In the case of degree 2 K3 surface, the relationship between various compactifications is well understood by work of Shah, Looijenga, and others. I will report on work in progress with K. O’Grady which aims to give similar complete description for degree 4 K3s.

Jack Hall (ANU)
January 30, 2015, 3:30-4:30

A generalization of Luna’s etale slice theorem

Let $X$ be an affine $\mathbb{C}$-variety with an action of a reductive group $G$. A beautiful theorem of Luna is that if $x\in X(\mathbb{C})$ has closed $G$-orbit, then there is a $G_x$-invariant and affine subvariety $W \hookrightarrow X$ containing $x$, where $G_x$ is the stabilizer of $x$, such that the induced map $W\times^{G_x} G \to X$ is $G$-etale. This result is very useful in applications. Unfortunately, direct extensions of Luna’s result to non-affine, singular varieties are not possible. By slightly weakening the conclusion, however, I will describe a generalization of Luna’s etale slice theorem covering these situations, and many more. This is joint work with Alper (ANU) and Rydh (KTH).

Alessandro Chiodo (Jussieu)
February 13, 2015

Néron models of Picard groups by Picard groups

The Néron model provides a universal extension over a discrete valuation ring R of the degree-zero part Pic0CK of the Picard group of a smooth curve CK over K=Frac(R). It is natural to exploit the (relative) Picard functor Pic0CR of a regular (semi)stable reduction CR to describe its Néron model N(Pic0CK). The group Pic0CR is not separated in general, but the Néron model N(Pic0CK) equals Pic0CR modulo the closure of the zero section of Pic0CK (Raynaud, 1970). In some very special cases, we obtain N(Pic0CK) without passing through the quotient by simply singling out the identity component of Pic0CR, that is the group of line bundles of degree 0 on all irreducible components. In general, the quotient does not possess a similar modular interpretation but this talk shows that the Néron model does represent a separated Picard functor of degree-0 line bundles on all irreducible components as soon as we adopt a stack-theoretic stable model of CK.

Daniel Murfet (USC)
February 20, 2015

Computing with hypersurfaces

Associated to an isolated hypersurface singularity is a triangulated category of matrix factorisations. A decade ago Khovanov and Rozansky figured out how to model knots in a bicategory built out of these triangulated categories. Inspired by their construction I will describe how matrix factorisations can be used to construct a model of the polymorphic lambda calculus of Girard, an abstract functional programming language which underlies, for example, Haskell. Beyond the sheer amusement of representing programs by objects of triangulated categories, there are motivations from within logic and computer science that I will also discuss.
Of course I will explain all the logic/comp-sci terms!

Srinivas (Tata Institute for Fundamental Research)
February 27, 2015, 3:20-4:20

Etale motivic cohomology and algebraic cycles

This talk will report on joint work with A. Rosenschon. There are examples (which I’ll briefly discuss) showing that the torsion and co-torsion of Chow groups are complicated, in general, except in the “classical” cases (divisors and 0-cycles, and torsion in codimension 2); also the integral Hodge conjecture is known to fail. Instead, we may (following Lichtenbaum)
consider the etale Chow groups, which coincide with the usual ones if we use rational coefficients; we show that they have better “integral” properties if we work over the complex numbers. In contrast, they can have infinite torsion in some arithmetic situations (the usual Chow groups are conjectured to be finitely generated).

Sándor Kovács (Washington)
February 27, 2015, 4:30-5:30

Projectivity of the moduli space of stable log-varieties

Stable log-varieties are the higher dimensional analogues of stable pointed curves and their moduli spaces are generalizations of $\bar M_{g,n}$. Just as in the curve case, extending the moduli problem to include stable objects leads to a compactification of the moduli space of the smooth ones. (These are known facts due to work of Koll​ár, Viehweg, Koll​ár-Shepherd-Barron, Alexeev, Keel-Mori, Hacon-McKernan-Xu and others). In the higher dimensional case there are many issues that come up that do not appear in dimension one. Furthermore, the log case is far from a straightforward generalization of the absolute case. In particular, Koll​ár’s powerful Ampleness Lemma does not apply in the log case. This talk is an overview report on recent and ongoing work, joint with Zsolt Patakfalvi, in which we prove a number of results: a generalization of Koll​ár’s Ampleness Lemma that works in the log case, various positivity results on pushforwards of relative pluricanonical sheaves and their determinants and as applications of these we prove that any complete moduli space of stable log-varieties is projective and we establish log-subadditvity of Kodaira dimension of log canonical fiber spaces.

Jie Zhou (Perimeter Institute)
April 3, 2015

Counting higher genus curves in local $\mathbf{P}^2$ via mirror symmetry

I will talk about the modularity of the generating series of Gromov-Witten invariants of local $\mathbf{P}^2$ by using mirror symmetry.

The moduli space involved in the mirror side of local $\mathbf{P}^2$ is naturally identified with a modular curve of genus zero. This implies that the mirror of the generating series, as sections of line bundles over the modular curve, are modular forms. The Fricke involution acting on the modular curve exchanges the asymptotic behaviour at the cusps of the modular forms. It induces a duality of the underling physics theories, in a way similar to the electro-magnetic duality in Seiberg-Witten theory.

The talk is based on a joint work with M. Alim, E. Scheidegger and S.-T. Yau.

Y.P. Lee (Utah)
April 10, 2015

A + B model in conifold transitions for Calabi–Yau threefolds

Witten named two topological field theories for Calabi–Yau threefolds A and B models respectively in his explanation of mirror symmetry. In this talk, A model is the Gromov–Witten theory and B model the variation of Hodge structure. All known examples of (simply-connected) Calabi–Yau threefolds are connected by a special kind of surgery, called (extremal) transition, of which the basic case is the conifold transition.

In this talk, I will explain a phenomenon of partial exchange of A and B models when the Calabi–Yau threefold undergoes a conifold transition. This suggests the possibility of an A+B theory which is invariant under transitions and is therefore equivalent for all (simply-connected) Calabi–Yau threefolds.

This talk is based on joint work with H.-W. Lin and C.-L. Wang.

Anand Patel (Boston College)
April 17, 2015

Algebraic cohomology of Hurwitz spaces

I will introduce the general problem of determining Chow rings of the Hurwitz spaces parametrizing branched covers of the projective line. I will present some conjectures as well as recent progress in the degree 3 case (joint with R. Vakil). This talk may be of interest to topologists as well – I will discuss ongoing work on the stabilization of the algebraic cohomology of Hurwitz spaces as the degree and genus parameters tend to infinity.

Simion Filip (Chicago)
May 1, 2015

Hodge theory and arithmetic in Teichmuller dynamics

The dynamics of a billiard ball in a polygon is a classical dynamical system for which many questions remain open. These questions are closely related to a natural action of the group SL(2,R) on the tangent bundle to the moduli space of Riemann surfaces. It can be viewed as a “complexified” geodesic flow. By recent results of Eskin and Mirzakhani, this action of SL(2,R) enjoys rigidity properties akin to Ratner’s theorems – in particular, orbit closures are submanifolds. In this talk, I will explain why these orbit closures are in fact algebraic varieties with interesting arithmetic properties. For instance, they parametrize algebraic curves with real multiplication and torsion conditions on (factors of) their Jacobian. These results depend on extending results about variations of Hodge structures to this special setting (in particular, giving a different approach to some results of Schmid). No background in any of the above topics will be assumed and I will provide the necessary introduction.

Monday May 4, 2015

Specialization of Quintic Threefolds to the Secant Variety

We consider the degeneration of quintic threefolds obtained by running semistable reduction on a general pencil of quintics specializing to the secant variety of a normal elliptic curve in projective 4-space. The question we would like to address is: what are the flat limits of the rational curves in this degeneration? In this talk, I will provide a good “first approximation” to the answer by describing the space of genus zero stable morphisms to the central fiber (as defined by J. Li).

Joe Rabinoff (Georgia Tech)
May 8, 2015

Integration on wide opens and uniform Manin-Mumford

Coleman’s effective version of Chabauty’s method of attacking the Mordell conjecture involves counting zeros of certain p-adic integrals on p-adic open discs, using a Newton polygon argument. Recently, Stoll extended the Chabauty–Coleman method using integration on open discs as well as open annuli, to prove a uniform Mordell conjecture for hyperelliptic curves of fixed genus and small Mordell–Weil rank. We use integration on so-called basic wide open subdomains, the p-adic analogue of a “pair of pants” in a Riemann surface, in order to prove a uniform Manin–Mumford result for curves of a fixed genus with sufficiently degenerate reduction type. This variant of the Chabauty–Coleman method is unique as it can be used to bound *geometric* points on a curve, not just rational points. Other ingredients include potential theory on Berkovich curves, the theory of linear systems on metric graphs, and a comparison of different p-adic integration theories. A special case of this method extends Stoll’s results to all curves of low Mordell–Weil rank.

This work is joint with Eric Katz and David Zureick-Brown. It complements David’s talk on April 13, but it is self-contained and assumes no prior knowledge of Berkovich spaces or the Chabauty–Coleman method.

Andrei Caldararu (Wisconsin)
May 15, 2015

Algebraic proofs of degenerations of Hodge-de Rham complexes

In the first half of the talk I shall present a new algebraic proof of a result of Deligne-Illusie about the degeneration of the Hodge-de Rham spectral sequence. The idea is to reduce the main technical point of their proof to a question about the formality of a derived intersection in an Azumaya space.

In the second half of the talk I shall discuss the main technical difficulty that arises when trying to extend these techniques to obtain a proof of a famous claim of Barannikov-Kontsevich. This claim, which was first proved by analytic methods by Sabbah, expresses the hypercohomology of the twisted de Rham complex in terms of computations with coherent sheaves. It is conceptually the analogue of the Hodge-de Rham degeneration statement for dg categories of matrix factorizations.

This is joint work with Dima Arinkin and Marton Hablicsek.

Alex Perry (Harvard)
May 29, 2015

Derived categories of Gushel–Mukai varieties

A Fano Gushel–Mukai (GM) variety is a Fano variety of Picard number 1, degree 10, and coindex 3. For such a variety $X$, there is a special subcategory $A_X$ of its derived category, which appears to be closely related to its birational geometry. I will discuss a duality operation that exchanges GM varieties of different dimensions and identifies their $A_X$ categories. As a special case, this gives a family of rational GM fourfolds whose category $A_X$ is equivalent to the derived category of a K3 surface. I will also discuss a relation between the derived categories of GM and cubic fourfolds. This is joint work with Alexander Kuznetsov.

Tom Coates (Imperial College London)
June 1, 2015, 4 pm (special algebraic and symplectic geometry seminar)

Mirror Symmetry and Fano Manifolds

Fano manifolds are basic building blocks in algebraic geometry, and the classification of Fano manifolds is a long-standing and important open problem. We explain a surprising connection between Mirror Symmetry and Fano classification. This is joint work with Akhtar, Corti, Galkin, Golyshev, Heuberger, Kasprzyk, Oneto, Petracci, Prince, and Tveiten.

Posted in Uncategorized

## stanford algebraic geometry seminar 2014-15

Winter and spring Fridays 3:45-4:45 in 383-N

Fall:  Fridays 4:30-5:30 in 383-N (with exceptions)

 September 26 at 4 pm Jarod Alper (ANU) Associated forms in classical invariant theory October 3 no seminar October 10 no seminar (WAGS weekend) October 11-12 Western Algebraic Geometry Symposium (University of Idaho) Christine Berkesch Zamaere, Jim Carlson , Giulio Caviglia, Dusty Ross, Karl Schwede, Sofia Tirabassi October 17 no seminar October 24 at 4:15 pm Jim Bryan (UBC) Donaldson-Thomas theory of local elliptic surfaces via the topological vertex October 31 at 4:30 pm Nasko Atanasov (Harvard) Interpolation and vector bundles on curves November 7 at 4:30 pm Yefeng Shen (Stanford) WDVV equations and Ramanujan identities November 14 at 4:30 pm Yiwei Shi (Chicago) The Shafarevich conjecture for K3 surfaces November 21 at 4:30 pm Sabin Cautis (UBC) Categorical Heisenberg actions on Hilbert schemes of points November 28 no seminar (Thanksgiving break) December 5 Francois Charles (Paris-Sud and MIT) Geometric boundedness results for K3 surfaces January 9 Dan Edidin (Missouri) Strong regular embeddings and the geometry of hypertoric stacks January 16 Evan O’Dorney (Harvard) Canonical rings of $\mathbf{Q}$-divisors on $\mathbf{P}^1$ January 23 Mark Shoemaker (Utah) A proof of the LG/CY correspondence via the crepant resolution conjecture January 30, 3:20-4:20 Radu Laza (Stony Brook) Moduli of degree 4 K3 surfaces revisited January 30, 4:30-5:30 Jack Hall (ANU) A generalization of Luna’s etale slice theorem February 13 Alessandro Chiodo (Jussieu) Néron models of Picard groups by Picard groups February 14-15 conference on “Moduli spaces of curves and maps” (more details later) February 20 Daniel Murfet (USC) Computing with hypersurfaces February 27, 3:20-4:20 Srinivas (TIFR) Etale motivic cohomology and algebraic cycles February 27, 4:30-5:30 Sándor Kovács (Washington) Projectivity of the moduli space of stable log-varieties February 28 – March 1 Western Algebraic Geometry Symposium (at UC Davis) April 3 Jie Zhou Counting higher genus curves in local $\mathbf{P}^2$ via mirror symmetry April 10 Y.P. Lee (Utah) A + B model in conifold transitions for Calabi–Yau threefolds April 17 Anand Patel (Boston College) Algebraic cohomology of Hurwitz spaces May 1 Simion Filip (U Chicago) Hodge theory and arithmetic in Teichmuller dynamics Monday May 4, 5 pm, 384-H (Student Algebraic Geometry Seminar) Adrian Zahariuc (Harvard) Specialization of Quintic Threefolds to the Secant Variety May 8 Joe Rabinoff (Georgia Tech) Integration on wide opens and uniform Manin–Mumford May 15 Andrei Caldararu (Wisconsin) Algebraic proofs of degenerations of Hodge-de Rham complexes May 22 no seminar (slot used by visitor of Yasha Eliashberg) May 29 Alex Perry (Harvard) Derived categories of Gushel–Mukai varieties Monday June 1, 4 pm, 383-N (joint with symplectic geometry seminar) Tom Coates (Imperial College London) Mirror Symmetry and Fano Manifolds

(Later talks: Tony Pantev (Penn), October 2.)

Posted in seminars

## abstracts for 2013-14 seminars

(The seminar webpage is here.)

Alexei Oblomkov (U Mass Amherst)
October 4, 2013

Plane curve singularities, knot homology and Hilbert scheme of points on plane

I will present a conjectural formula for the Poincare polynomial of the Hilbert scheme of points on a planar curve (joint with Rasmussen and Shende).  The formula is written in terms of the Khovanov-Rozansky invariants of the links of the singularities of the curve.  In the case of toric curve $\{ x^m=y^n\}$ the Poincare polynomial also could be written in terms of equivariant Euler characteristic of some sheaf of some particular equivariant sheaf on $\text{Hilb}^n (\mathbb{C}^2)$ (joint with Yun). If time permits I will also discuss cohomology ring of the compactified Jacobian of the toric curve and the conjectural description of the ring for general curve singularity (joint with Yun).

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Francois Greer (Stanford)
October 25, 2013

Picard Groups of Moduli Spaces of K3 Surfaces

Polarized K3 surfaces of genus $g$ can be thought of as families of canonical curves.  As such, their moduli space $K_g$ has similar properties to $M_g$.  For instance, both are unirational for low values of g, and both have discrete Picard group.  In this talk, we will use the explicit unirationality of $K_g$ in the range of Mukai models to compute its Picard number, and verify the Noether-Lefschetz conjecture for genus up to 10.  This is joint work with Zhiyuan Li and Zhiyu Tian.

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Bhargav Bhatt (IAS)
November 1, 2013

Lefschetz for local Picard groups

A classical theorem of Lefschetz asserts that non-trivial line bundles on a smooth projective variety of dimension $\geq 3$ remain non-trivial upon restriction to an ample divisor. In SGA2, Grothendieck recast this result in purely local algebraic terms. Answering a question raised recently by Koll\’ar, we will explain how this local reformulation remains true under milder hypotheses than those imposed in SGA2. Our approach relies on a vanishing theorem in characteristic p, and formal geometry over certain very large (non-noetherian) schemes. This is joint work with Johan de Jong.

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Mark de Cataldo (Stony Brook University)
November 15, 2013

The projectors of the decomposition theorem are absolute Hodge

I report on joint work with Luca Migliorini at Bologna. If you have a map of complex projective manifolds, then the rational cohomology of the domain splits into a direct sum of pieces in a way dictated by the singularities of the map. By Poincaré duality, the corresponding projections can be viewed as cohomology classes (projectors) on the self-product of the domain. These projectors are Hodge classes, i.e. rational and of type (p,p) for the Hodge decomposition. Take the same situation after application of an automorphism of the ground field of complex numbers. The new projectors are of course Hodge classes. On the other hand, you can also transplant, using the field automorphism, the old projectors into the new situation and it is not clear that the new projectors and the transplants of the old projectors coincide. We prove they do, thus proving that the projectors are absolute Hodge classes, i.e. their being of Hodge type survives the totally discontinuous process of a field automorphism.  We also prove that these projectors are motivated in the sense of Andre.

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Allen Knutson (Cornell)
November 22, 2013

Conormal varieties and the Temperley-Lieb algebra

Each permutation pi has an associated determinantal variety called its “matrix Schubert variety”. If one degenerates these determinants to monomials, the components of the resulting scheme are naturally indexed by reduced words for pi as a product of simple reflections ([K-Miller 2005]). It is particularly notable that unlike in most limits of this sort, the resulting scheme has no multiplicities on its components.

I’ll describe the “conormal variety” to a variety, which shows up in many contexts (e.g. projective duality), and extend this degeneration idea to the conormal varieties of the matrix Schubert varieties. The limit scheme now includes the conormal variety to the original limit, and to its projective dual, but also some fundamentally new components which appear with multiplicity. Then I’ll state some conjectures, in particular that the components are naturally indexed by words in the generators of the Temperley-Lieb algebra, from which one can also predict the multiplicities.

This work is joint with Paul Zinn-Justin.

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Yuri Tschinkel (Courant (=NYU) and the Simons Foundation)
Monday, December 2, 2013, 2:30 pm (note unusual date and time)

Igusa integrals

Geometric Igusa integrals appear as important technical tools in the study of rational and integral points on algebraic varieties. I will describe some of these applications (joint work with A. Chambert-Loir).

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Nick Katz (Princeton)
January 31, 2014

Equidistribution questions arising from universal extensions

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Akhil Mathew (Harvard)
Tuesday February 18, 4 pm, 383-N (joint with topology)

The Galois group of a stable homotopy theory

To a “stable homotopy theory” (a presentable, symmetric monoidal stable ∞-category), we naturally associate a category of finite ́etale algebra objects and, using Grothendieck’s categorical machine, a profinite group that we call the Galois group. This construction builds on, and generalizes, ideas of many authors, and includes the ́etale fundamental group of algebraic geometry as a special case. We calculate the Galois groups in several examples, both in settings of rational and p-adic homotopy and in “chromatic” stable homotopy theories. For instance, we show that the Galois group of the periodic E∞-algebra of topological modular forms is trivial, and, extending work of Baker and Richter, that the Galois group of K(n)-local stable homotopy theory is an extended version of the Morava stabilizer group.

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Bernd Sturmfels (Berkeley)
February 28, 2014

The Euclidean Distance Degree of an Algebraic Variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points of this optimization problem. We focus on varieties encountered in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

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Kiran Kedlaya (UCSD)
March 7, 2014

Sato-Tate groups of abelian surfaces

There are a number of theorems and conjectures in arithmetic geometry which control the structure of an abelian variety over a number field in terms of its reductions to various finite fields. Here we consider a rather crude question along these lines: how much structure can be read off by computing the zeta functions of these reductions and retaining only the statistical behavior of these zeta functions as one averages over primes? For elliptic curves, this is known (under appropriate conjectures, which are known in many cases) to distinguish whether or not the curve has complex multiplication, and if so whether this happens over the base field or an extension field. We describe the corresponding picture for genus 2 curves: the punchline is that there are 52 different possible cases, corresponding to Galois module structures on real endomorphism algebras. These cases will be illustrated with some pretty pictures! Joint work with Francesc Fite, Victor Rotger, and Andrew Sutherland.

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June Huh (Michigan)
May 2, 2014

Homology classes in algebraic varieties: nef, effective, and prime

The homology group of an algebraic variety is an abelian group equipped with several additional structures. It contains the set of primes, the homology classes of subvarieties. It contains the semigroup of effective classes, the nonnegative linear combinations of primes. It contains the semigroup of nef classes, the classes which intersect all primes of complementary dimension nonnegatively. We will see how these subsets look like in a particular algebraic variety, the one associated to the polytope permutahedron’. The semigroups in this case have tractable structures, while the distribution of primes is more mysterious and related to some deep combinatorial conjectures on matroids.

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Zhiyuan Li (Stanford)
May 9, 2014

Special cycles on Shimura varieties of orthogonal type

Let $Y$ be a connected smooth Shimura variety. There are many special algebraic cycles coming from sub-Shimura variety of the same type in all codimensions, called special cycles on $Y$. A Hodge type question is whether these special cycles exhaust all the low degree cohomology classes of $Y$. In this talk, I will discuss relation between this question and Arthur’s theory and briefly talk about the work of Beregon-Millson-Moeglin in this direction.

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Qile Chen (Columbia)
May 16, 2014

$\mathbb{A}^1$-curves on quasi-projective varieties

The theory of stable log maps was developed recently for studying the degeneration of Gromov-Witten invariants. In this talk, I will introduce another interesting aspect of stable log maps as a useful tool for investigating ${\mathbb{A}}^1$-curves on quasi-projective varieties, which are the analogue of rational curves on proper varieties. At least two applications of ${\mathbb{A}}^1$-curves will be discussed in this talk. For classical birational geometry, the ${\mathbb{A}}^1$-curves can be used to produce very free rational curves on general Fano complete intersections in projective spaces. On the arithmetic side, ${\mathbb{A}}^1$-connectedness gives a general frame work for the existence of integral points over function field of curves. This is a joint work in progress with Yi Zhu.

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Aaron Pixton (Clay)
May 23, 2014

Double ramification cycles and tautological relations

Double ramification cycles parametrize curves that admit maps to the projective line with specified ramification over zero and infinity. They can be extended to the moduli space of stable curves by using the virtual class in relative Gromov-Witten theory. I will describe a conjectural formula for these extensions in terms of tautological classes. The formula is motivated by a connection with recent joint work with Pandharipande and Zvonkine on Witten’s r-spin class, and it comes with a family of tautological relations which extend relations studied by Grushevsky and Zakharov.

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John Lesieutre (MIT)
May 30, 2014

Negative answers to some positivity questions

I will explain how the classically-studied action of Cremona transformations on configurations of points in projective space can be used to construct several counterexamples in birational geometry: nefness is not an open condition in families, Zariski decompositions do not always exist in dimension 3, and a variety can have infinitely many Fourier-Mukai partners. As time permits, I’ll discuss some related examples on Calabi-Yau threefolds.

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Posted in Uncategorized