Harvard/MIT Algebraic Geometry Seminar
Spring 2017
Tuesdays at 3 pm
The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT
(4153) and Harvard (Science Center 507).

Feb 72017
SC 507Harvard
Aaron Bertram, University of Utah
Counting finite quot schemes
abstract±
Grothendieck's quot scheme generalizes the Grassmannian (the variety of quotients of a vector space) to produce a projective scheme of quotients of a fixed coherent sheaf on a fixed variety. When the variety is a curve or del Pezzo surface, there are instances in which the quot scheme is both of expected dimension zero and actually of dimension zero, in which case we may ask for its cardinality. Over a curve, the cardinality of these quot schemes is the trace of a matrix, coming from a topological quantum field theory that contains both the quantum cohomology of the Grassmannian and the Verlinde formula. In the surface case, the cardinality is much more mysterious. This is work by my student Thomas Goller, and also joint work with with Thomas and Drew Johnson.

Feb 142017
4153MIT
Arnav Tripathy
Further counterexamples to the integral Hodge conjecture
abstract±
I'll briefly recall the history of counterexamples to the integral Hodge conjecture, starting with the topological motivations of Atiyah and Hirzebruch. I'll then transition to describing a new class of counterexamples to the integral Hodge and integral Tate conjectures that I show exist according to a conjecture of Ben Antieau.

Feb 212017
SC 507Harvard
David Hyeon, Seoul National University
Commuting nilpotents modulo simultaneous conjugation and Hilbert scheme
abstract±
Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers. But the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all although it has a definite moduli theory flavor. Unlike the case of commuting nilpotents paired with a cyclic vector,
the GIT is not well behaved in this case. I will explain how a 'moduli space' can be constructed as a homogeneous space, and show that it is isomorphic to an open subscheme of a punctual Hilbert scheme. Over the field of complex numbers, thus constructed space is diffeomorphic to a direct sum of twisted tangent bundles over a projective space. This is a joint work with W. Haboush.

Feb 282017
4153MIT
Daniel Litt, Columbia
Arithmetic Restrictions on Geometric Monodromy
abstract±
Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X.
As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) satisfying the following: any irreducible, nontrivial padic representation of the fundamental group of X, which arises from geometry, is nontrivial mod p^N.

Mar 72017
SC 507Harvard
Bill Fulton, U Michigan
Degeneracy Loci with a Line Bundle
abstract±
In order to give formulas for classical as well as modern
degeneracy loci, one needs to allow symplectic or
quadratic forms on a vector bundle V with values in a
line bundle L. If one has two flags of isotropic subbundles
of V, one has a degeneracy locus for each signed permutation,
specifying how they intersect. Finding formulas for their
cohomology classes amounts to constructing double Schubert
polynomials with a new parameter corresponding to the first
Chern class of L. We construct these “twisted double Schubert
polynomials” and show they satisfy all the analogues of the
type A polynomials, including positivity for their coefficients and
their products. This is joint work with Dave Anderson, answering
a question of Joe Harris from 1987.

Mar 142017
4153MIT
Aleksey Zinger, Stony Brook  RESCHEDULED TO MAY 16 DUE TO SNOW
Enumerative geometry of curves: old and new
abstract±
I will present an overview of the (complex) GromovWitten
invariants and their relation to curve counts provided by
Pandharipande's version of GopakumarVafa formula for Fano classes. I will then present a similar formula that transforms the real positivegenus GWinvariants of many realorientable threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. This talk is based on joint works with P. Georgieva and J. Niu.

Mar 212017
SC 507Harvard
Brian Osserman, UC Davis
Limit linear series and the maximal rank conjecture
abstract±
The maximal rank conjecture addresses the degrees of equations cutting
out suitably general curves in projective spaces. We describe an approach
to this conjecture involving degenerating to a chain of genus1 curves,
and using ideas from limit linear series. More speculatively, we will
describe the prospects for applications to related problems such as the
strong maximal rank conjecture. This is joint work with Fu Liu, Montserrat
Teixidor i Bigas, and Naizhen Zhang

Mar 282017
SC 507Harvard
David Stapleton
Hilbert schemes of points on surfaces and their tautological bundles
abstract±
Fogarty showed in the 1970s that the Hilbert scheme of n points on a smooth surface is itself smooth. Interest in these Hilbert schemes has grown since it has been shown they arise in hyperkahler geometry, geometric representation theory, and algebraic combinatorics. In this talk we will explore the geometry of certain tautological bundles on the Hilbert scheme of points. In particular we will show that these tautological bundles are (almost always) stable vector bundles. We will also show that each sufficiently positive vector bundles on a curve C is the pull back of a tautological bundle from an embedding of C into the Hilbert scheme of the projective plane.

Apr 42017
SC 507Harvard
Scott Mullane
Extremal effective divisors in M_{g,n}
abstract±
An abelian differential defines a flat metric with singularities at its zeros and poles, such that the underlying Riemann surface can be realised as a polygon whose edges are identified pairwise via translation. A number of questions about geometry and dynamics on Riemann surfaces reduce to studying the strata of abelian differentials with prescribed number and multiplicities of zeros and poles.
In this talk, we will focus on the divisorial strata closures that form special codimensionone subvarieties in the the DeligneMumford compactified moduli space of Riemann surfaces. For genus g>1 curves with n>g marked points, we show that infinitely many of these divisors form extremal rays of the cone of effective divisors. Hence these effective cones are not rational polyhedral.

Mon. Apr 102017
2361MIT
SPECIAL TALK 12 PM: Mihnea Popa, Northwestern
Families of varieties and Hodge theory
abstract±
Allowing the isomorphism class of algebraic varieties to vary in a family usually imposes strong conditions on the space parametrizing that family. This has been thoroughly studied for families of varieties whose canonical bundle is positive, leading to what is called the hyperbolicity of the moduli stack of such varieties. I will explain how recent advances in Hodge theory and in the study of holomorphic forms, facilitated especially by M. Saito's theory of Hodge modules, allow us to answer wellknown questions regarding the base spaces of more general (and conjecturally arbitrary) families of varieties. Most of this is joint work with C. Schnell.

Apr 112017
4153MIT
Mihnea Popa, Northwestern
Hodge ideals
abstract±
I will present joint work with M. Mustata, in which we study a sequence of ideals arising naturally from M. Saito's Hodge filtration on the localization along a hypersurface. The multiplier ideal of the hypersurface appears as the first step in this sequence, which as a whole provides a more refined measure of singularities. We give applications to the comparison between the Hodge filtration and the pole order filtration, adjunction and restriction theorems, and the singularities of theta divisors on abelian varieties.

Fri. Apr 142017
5234MIT
Special Seminar at 1 PM: Andrei Okounkov
Geometric construction of integral solutions of qKZ equations
abstract±
The quantum KnizhnikZamolodchikov equations of I. Frenkel and N. Reshetikhin are flat qdifference connections that play a very important role in the representation theory of quantum affine Lie algebras and its application in mathematical physics. It is a wellknown and much studied problem to solve these equations by integrals. I will explain a geometric solution to this problem, in the generality of an arbitrary Lie algebra associated to a quiver, following a joint paper with Mina Aganagic.

Apr 182017
SC 507Harvard
Angelo Vistoli, Scuola Normale
Motivic classes of classifying spaces
abstract±
If G is an affine group over a field k, BehrendDhillon and Ekedahl have defined the class {BG} of the classifying space BG in an appropriate localization of the Grothendieck ring of varieties. I will discuss the little that is known about these classes, and the suggestive, if completely hypothetical, connection with the problem of rationality of fields of invariants.

Apr 252017
SC 507Harvard
Yaim Cooper, Harvard
A Fock Space Approach to Severi Degrees
abstract±
I will describe an approach to curve counting on some rational surfaces such as P^{2} and Hirzebruch surfaces using operators on a Fock space. As a consequence, we obtain formulas comparing GromovWitten invariants and enumerative curve counting on these surfaces, extending a result of Abramovich and Bertram comparing enumerative curve counts on F_{0} and F_{2} to F_{1} and F_{3}. We also find a pair of differential equations satisfied by relative GromovWitten invariants of F_{k}, one of which recovers a formula of Getzler and Vakil. This work is partly joint with R. Pandharipande.

May 22017
4153MIT
Yongbin Ruan, U Michigan
Computing higher genus GromovWitten of the quintic
abstract±
In every subject, there are a few guiding problems
which are usually very difficult and yet inspire much of activities.
The computing higher genus GromovWitten invariants of quintic
(more generally compact CalabiYau manifolds) is such an example.
I have been engaging this problem for a large part of past fifteen
years. Sometimes, I felt close and even had a bet with physicist Klemm
several years ago. Some other times, the resolution of the problem
seems to be far off. Nevertheless, it inspired the much developments
such as the modularity properties, FJRWtheory and recent mathematical
GLSMtheory. In the talk, I will share with audience my journey during last
fifteen years and some recent outlook.

May 162017
4153MIT
Aleksey Zinger, Stony Brook
Enumerative geometry of curves: old and new
abstract±
I will present an overview of the (complex) GromovWitten
invariants and their relation to curve counts provided by
Pandharipande's version of GopakumarVafa formula for Fano classes. I will then present a similar formula that transforms the real positivegenus GWinvariants of many realorientable threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. This talk is based on joint works with P. Georgieva and J. Niu.
This seminar is organized by Joe Harris (Harvard), Bjorn Poonen (MIT), Davesh Maulik (MIT), Sam Raskin (MIT), Maksym Fedorchuk (BC), Dawei Chen (BC), Yaim Cooper
(Harvard), Anand Patel (BC), Aaron Pixton (Harvard). This seminar is supported in part by
grants from the NSF. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s)
and do not necessarily reflect the views of the National Science
Foundation.