Harvard/MIT Algebraic Geometry Seminar
Spring 2018
Tuesdays at 3 pm
The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT
(4153) and Harvard (Science Center 507).

Feb 62018
4153 MIT
Bhargav Bhatt, Michigan
Prisms and deformations of de Rham cohomology
abstract±
In arithmetic geometry, there are multiple instances where the de Rham cohomology of a smooth variety admits a natural deformation (such as crystalline cohomology or the recently constructed A_{inf}cohomology). I will explain a general sitetheoretic approach, relying on a notion we call prisms, that produces such deformations. In addition to recovering the known deformations in a new and simple way, this framework also constructs some previously conjectural ones (qdeformations of de Rham cohomology, cohomological BreuilKisin or Wach modules). This is a report on work in progress with Peter Scholze.

Feb 132018
SC 507 Harvard
Jesse Kass, University of South Carolina
How to count lines on a cubic surface arithmetically
abstract±
Salmon and Cayley proved the celebrated 19th century result that a smooth cubic surface over the complex numbers contains exactly 27 lines. By contrast, the count over the real numbers depends on the surface, and these possible counts were classified by Segre. Benedetti–Silhol, Finashin–Kharlamov and Okonek––Teleman made the striking observation that Segre’s work shows a certain signed count is always 3. In my talk, I will explain how to extend this result to an arbitrary field. Although I will not use any homotopy, I will draw motivation from A1homotopy theory. This is joint work with Kirsten Wickelgren.

Feb 202018
4153 MIT
Angela Gibney, Rutgers
Vector bundles of conformal blocks on the moduli space of curves
abstract±
In this talk I will give a tour of recent results and open problems about vector bundles of conformal blocks on the moduli space of curves. I will discuss how these results fit into the context of some of the open problems about the birational geometry of the moduli space.

Feb 272018
SC 507Harvard
Nick Salter, Harvard
Vanishing cycles for linear systems on toric surfaces
abstract±
Given a linear system on a smooth complex toric surface (eg the projective plane or P^1 \times P^1), it is very natural to ask about the set of all possible vanishing cycles. That is, for a fixed identification of a smooth fiber with a topological surface, which simple closed curves can be shrunk to a point via a nodal degeneration of curves in the linear system? Recent work of mine gives a complete answer to this question. It turns out that the essential invariant that determines whether a simple closed curve is a vanishing cycle is an ``rspin structure’’ of the sort studied by Witten and others in the context of cohomological field theory. The techniques are essentially topological, and are based on a reformulation of the problem in terms of the mapping class groupvalued monodromy representation of the linear system.

Mar 62018
4153MIT
Michael Kemeny, Stanford
Betti numbers of the canonical ring of a curve
abstract±
Understanding the structure of the canonical ring of a curve has long been of fundamental interest
in algebraic geometry. This structure can be encoded in certain invariants called the "graded Betti numbers".
A conjecture of Mark Green (now essentially a theorem of Voisin) predicts exactly which of these invariants vanish,
but little has been known about the values of those Betti numbers which do not vanish, and what geometric information
they might contain. We will explain an approach to these problems using Hurwitz spaces and state a result proving
that the "extremal" Betti number counts minimal pencils, modulo transversality hypotheses (verifying a philosophy of Schreyer).

Mar 14, 34PM 2018
2449 MIT
Ariyan Javanpeykar, Mainz (NOTE ROOM/TIME CHANGE)
Arithmetic, algebraic, and analytic hyperbolicity
abstract±
Abstract. In the first part of this talk we will discuss different notions of hyperbolicity for algebraic varieties which are conjecturally related by conjectures of
GreenGriffiths, Lang, and Vojta. Then, we introduce an "analytic" notion of hyperbolicity, which interpolates between being Brody hyperbolic and
hyperbolically embedabble. I hope to give a survey of several known results in the first part. In the second part I will focus on arithmetic questions. For instance, Scholl proved that the
moduli of del Pezzo surfaces is arithmetically hyperbolic. In joint work with Daniel Loughran, we investigate the analogous question for Fano threefolds. For instance, we show that the
moduli of Fano threefolds is not arithmetically hyperbolic, by (explicitly) constructing an abelian surface which sits inside this moduli.

Mar 202018
4153MIT
Tathagata Basak, Iowa State
A complex ball quotient and the monster
abstract±
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 BarnesWall lattice for which there is a similar story.

Mar 272018
SC 507Harvard
Allen Knutson, Cornell
Deformations of normal crossings
abstract±
The standard normal crossings divisor on A^n is given by the vanishing of the degree n polynomial prod_i(x_i). Let p be a polynomial with this as leading term, and consider the stratification it "generates" (by taking components, intersections thereof, and iterating). I'll prove some very nice properties of this stratification (via char p techniques), and show that Bruhat cells naturally come with such good stratifications. I'll explain a sense in which the Bruhat stratifications form an order ideal in the family of all such stratifications, and give a natural example that isn't itself Bruhat. Finally, I'll give examples of stratified projective varieties covered with charts isomorphic to Bruhat cells, such as Grassmannians and wonderful compactifications of groups.

Apr 32018
SC 507Harvard
Curt McMullen, Harvard
Billiards, quadrilaterals and moduli spaces
abstract±

Apr 102018
SC 507Harvard
Eric Larson, MIT
The Maximal Rank Conjecture
abstract±
Let C be a general curve of genus g, embedded via a general
linear series of degree d in P^r. In this talk, we determine the
Hilbert function of C.

Apr 172018
4153MIT
June Park, Brown University
Arithmetic of the moduli of quasimaps and the moduli of fibered algebraic surfaces with heuristics for counting curves over global fields
abstract±
We give enumerative study of the moduli of quasimaps in terms of the motive counts in Grothendieck ring of variety and relate it to the moduli of genus 0 twisted maps. As a corollary, we acquire the arithmetic of the moduli of elliptic surfaces over P^1. We also provide the family of asymptotics for the point counts of the moduli of genus g hyperelliptic fibrations over P^1 focusing upon the genus 2 case. In the end, we pass the acquired arithmetic invariants through the global fields analogy which renders new heuristics of Z_g(B) for counting the elliptic or hyperelliptic genus g curves over Q by the bounded height of discriminant. This is a report on work in progress with Kenneth Ascher, Dori Bejleri and Changho Han.

Apr 242018
SC 507Harvard
Melody Chan, Brown
Cohomology of M_g and the tropical moduli space of curves
abstract±
Joint work with Søren Galatius and Sam Payne. We study the rational
cohomology of M_g in degree 4g6, showing that it grows at least
exponentially in g. I will explain the relationship with the tropical
moduli space of curves, and the combinatorial topology behind our
result.

May 12018
4153MIT
BATMOBYLE@MIT
http://math.mit.edu/~kascher/BATMOBYLE/spring2018.html
abstract±

May 82018
SC 507Harvard
Renzo Cavalieri, Colorado State
Witten conjecture for Mumford's kappa classes
abstract±
Kappa classes were introduced by Mumford, as a tool to explore the intersection theory of the moduli space of curves. Iterated use of the projection formula shows there is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves, and the intersection theory of psi classes on all moduli spaces. In terms of generating functions, we show that the potential for kappa classes is related to the GromovWitten potential of a point via a change of variables essentially given by complete symmetric polynomials, rediscovering a theorem of Manin and Zokgraf from '99. Surprisingly, the starting point of our story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm (the relevant graphs being dual graphs to stable curves).This is joint work (in progress) with Vance Blankers.
This seminar is organized by Joe Harris (Harvard), Davesh Maulik (MIT), Brooke Ullery (Harvard), Philip Engel (Harvard), Dhruv Ranganathan (MIT), Rohini Ramadas (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.