Harvard/MIT Algebraic Geometry Seminar
Fall 2018
Tuesdays at 3 pm
The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT
(2142) and Harvard (Science Center 507).
*****We are no longer going by Harvard time. The seminar will start at 3 sharp!*****

Sept 112018
SC 507 Harvard
Jack Huizenga, Penn State
Moduli of sheaves on Hirzebruch surfaces
abstract±
Let X be a Hirzebruch surface. Moduli spaces of semistable sheaves on X with fixed numerical invariants are always irreducible by a theorem of Walter. On the other hand, many other basic properties of sheaves on Hirzebruch surfaces are unknown. I will discuss two different problems on this topic. First, what is the cohomology of a general sheaf on X with fixed numerical invariants? Second, when is the moduli space actually nonempty? The latter question should have an answer reminiscient of the DrezetLe Potier classification of semistable sheaves on the projective plane; in particular, there is a fractallike hypersurface in the space of numerical invariants which bounds the invariants of semistable sheaves. This is joint work with Izzet Coskun.

Sept 182018
2142 MIT
Austin Conner, Texas A&M
New approaches to upper bounds on the complexity of matrix multiplication
abstract±
I will describe two new paths using algebraic geometry and representation theory to prove upper bounds on the exponent of matrix multiplication. The first approach aims to apply the laser method of Strassen to previously unstudied tensors which uniquely share certain geometric properties with the CoppersmithWinograd tensor. The second approach for upper bounds transforms the problem into that of finding a certain sequence of finite groups and associated representations. The first approach is joint work with Fulvio Gesmundo, JM Landsberg, and Emanuele Ventura.

Sept 252018
SC 507 Harvard
Junliang Shen, MIT (We're starting at 3 pm sharp from now on)
Hilbert schemes and the P=W conjecture for parabolic Higgs bundles
abstract±
The P=W conjecture by de CataldoHauselMigliorini establishes a surprising connection between
the topology of the Hitchin system for the moduli of Higgs bundle and the mixed Hodge structure for the
corresponding character variety with respect to Simpson's nonabelian Hodge theorem.
We will present a proof of the P=W conjecture for the moduli spaces of parabolic Higgs bundles (of arbitrary rank)
associated to five affine Dynkin diagrams. Our proof relies on the study of the tautological classes and the perverse
filtration on the Hilbert scheme of points on surfaces. Based on joint work with Zili Zhang.

Oct 22018
2142 MIT
Tom Bachmann, MIT
Affine Grassmannians in motivic homotopy theory
abstract±
Motivic homotopy theory was invented by MorelVoevodsky in order to
rigorously apply homotopy theoretical ideas to the study of schemes. In
particular, for every field k they construct a closed symmetric monoidal
category H(k)_* called the homotopy category of pointed motivic spaces
over k, and every pointed smooth kvariety X has a homotopy type t(X) in
H(k)_*. On the other hand if G is a group scheme over k, then the affine
Grassmannian Gr(G) is an indscheme devised as an algebraic analog of
the loop space of G and of importance in geometric representation
theory. It is possible to make sense of t(Gr(G)) in H(k)_*. I will
explain the following result: t(Gr(G)) is obtained from t(G) by applying
the functor Hom(t(Gm), ), where Hom means internal hom and Gm denotes
the pointed scheme (A^1  0, 1). In other words in motivic homotopy
theory, affine grassmannians really are honest loop spaces.

Oct 92018
SC 507 Harvard
Ignacio Barros, Northeastern
Uniruledness of strata of holomorphic differentials in small genus
abstract±
We address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Of special interest is the case of genus ten where uniruledness is obtained via the study of the Severi variety of nodal curves on K3 surfaces. We also construct projective bundles over rational varieties that dominate the holomorphic strata with length at most g  1, hence showing in addition that these strata are unirational.

Oct 162018
2142 MIT
Dori Bejleri, MIT
Compact moduli of elliptic fibrations and degree one del Pezzo surfaces
abstract±
A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical CayleyBacharach 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 anticanonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs (X,D) is the natural extension to dimension 2 of the DeligneMumfordKnudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of spaces 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.

Oct 232018
SC 507 Harvard
Tim Magee, Instituto de Mathemáticas, UNAM
Toric degenerations of cluster varieties
abstract±
Cluster varieties are a particularly nice class of log CalabiYau varieties the noncompact analogue of usual CalabiYaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural progression from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations. And examples are abundant, with many applications in representation theory and Teichmüller theory.
Compactifications of A and their toric degenerations were studied extensively by Gross, Hacking, Keel, and Kontsevich. These compactifications generalize the polytope construction of toric varieties a construction which is recovered in the central fiber of the degeneration. Compactifications of X were introduced by Fock and Goncharov and generalize the fan construction of toric varieties. Recently, Lara Bossinger, Juan Bosco Frías Medina, Alfredo Nájera Chávez, and I introduced the notion of an Xvariety with coefficients, expanded upon the notion of compactified Xvarieties, and for each torus in the atlas gave a toric degeneration where each fiber is a compactified Xvariety with coefficients. We showed that these fibers are stratified, and each stratum is a union of compactified Xvarieties with coefficients. In the central fiber, we recover the toric variety associated to the fan in question, and we show that strata of the fibers degenerate to toric strata. This talk is based on arXiv:1809.08369 [math.AG].

Oct 2411noon
2449MIT
Harold Blum, Utah (NOTE UNUSUAL TIME/DAY/LOCATION)
Moduli of uniformly Kstable Fano varieties
abstract±
In order to have a well behaved moduli functor for Fano varieties, it seems natural to restrict oneself to Fano varieties that are Kstable. Recall, Kstability is an algebraic notion that characterizes when a smooth Fano variety admits a KahlerEinstein metric. In this talk, we consider the behavior of uniform Kstability (a strengthening of Kstability) in families. We will explain that uniform Kstability is an open condition in QFano families and the moduli functor of uniformly Kstable QFano varieties is separated. Together with a boundedness result of C. Jiang, these results yield a separated DM stack parameterizing uniformly Kstable Fano varieties of fixed dimension and volume. This is joint work with Yuchen Liu and Chenyang Xu.

Oct 302018
2142 MIT
John Calabrese, Rice
The crepant resolution conjecture in DonaldsonThomas theory
abstract±
DonaldsonThomas theory is a curvecounting theory closely related to the one of GromovWitten. I will discuss the proof of a comparison formula for DT invariants, which takes place in the context of the McKay correspondence. The formula was conjectured by BryanCadmanYoung, and the proof is work in progress joint with Beentjes and Rennemo.

Nov 62018
SC 507 Harvard
Alex Perry, Columbia
Stability conditions and cubic fourfolds
abstract±
Kuznetsov showed the derived category of a cubic fourfold contains a special subcategory which can be thought of as a "noncommutative K3 surface". The goal of this talk is to describe the structure of moduli spaces of Bridgeland stable objects on this noncommuative K3 surface, together with several applications. This is joint work with Bayer, Lahoz, Macri, Nuer, and Stellari.

Nov 132018
2142 MIT
Lynn Chua, UC Berkeley
Schottky Algorithms: Classical meets Tropical
abstract±
We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the SchottkyIgusa modular form. This is joint work with Mario Kummer and Bernd Sturmfels.

Nov 202018
SC 507 Harvard
Laure Flapan, Northeastern
Monodromy of Kodaira fibrations
abstract±
A longstanding question in studying the topology of complex algebraic varieties is the question of what groups can occur as the fundamental group of a smooth projective variety. One approximation of this question in the case of fibered varieties is to ask what groups can occur as the monodromy group of such a fibration. We use Hodge theory to investigate this question in the case of fibered algebraic surfaces, called Kodaira fibrations, whose fibers are all smooth and draw connections with questions about Shimura varieties and the moduli space of smooth algebraic curves.

Nov 272018
2142 MIT
Aaron Bertram, Utah
Stability for Regular Coherent Sheaves
abstract±
The Gieseker slope of a coherent sheaf on projective space is the ratio:
m(t) = Hilbert polynomial/Leading coefficient. This is a polynomial that is used to define
an open stability condition with (quasi)projective moduli. If, however, we consider
the rational function n(t) = Hilbert polynomial/Derivative, then we get a oneparameter family of open stability
conditions on sheaves (and complexes) of varying CastelnuovoMumfordregularity.
This produces moduli that exhibit (GIT) wallcrossing and converge (conjecturally) to the moduli spaces
of Giesekerstable sheaves. This recovers, for example, the construction by PieneSchlessinger of
the Hilbert scheme of twisted cubics, as was shown by Schmidt and Xia.
This is joint work with Altavilla, Mu and Petkovic.

Dec 42018
SC 507 Harvard
Rob Silversmith, Northeastern
Symmetric products of projective space and GromovWitten theory
abstract±
I'll discuss (orbifold) symmetric products of projective spaces Sym^d(P^r), focusing on the structure of orbits of the natural torus action. These orbits vary in nice (essentially toric) moduli spaces  for example, components of the moduli space of 1dimensional orbits are naturally identified with certain toric compactifications of M_{0,n}. I will then discuss how one can use these orbits to calculate GromovWitten invariants of Sym^d(P^r). If I have time, I'll also talk about how this story works when Sym^d(P^r) is replaced with the Hilbert scheme of points Hilb^d(P^r). This talk will be accessible to graduate students.

Dec 112018
2142 MIT
Dima Arinkin, Wisconsin
Cameral covers and Higgs bundles: additive, multiplicative, and elliptic.
abstract±
Higgs bundles are natural geometric objects that have been studied from many different directions. One of the key tools is the Hitchin fibration, which is the geometric version of the fundamental idea from linear algebra: the data (Higgs bundle) is split into spectral data ('eigenvalues') and spacial data (`eigenspaces'). A further development of this idea is the theory of cameral covers due to R.Donagi and D.Gaitsgory.
This seminar is organized by Joe Harris (Harvard), Davesh Maulik (MIT), Brooke Ullery (Harvard), Dhruv Ranganathan (MIT). 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.