Boston College 2021

Links: We will send out a Zoom link to all registered participants. Note that you must register to receive the link. There will also be a Gather.town session lasting all day Saturday (which will include the poster session).

Schedule: Each talk will last for 20 minutes, with 5 minutes for questions afterwards. All scheduled times are in the Eastern time zone.

Titles and abstracts:

Samir Canning (UCSD) + Hannah Larson (Stanford): Chow rings of Hurwitz spaces and moduli spaces of curves

We outline our results from a series of papers about the Chow rings of Hurwitz moduli spaces and the moduli spaces of curves. We will introduce the notion of tautological classes for both moduli spaces. We then will explain how our study of the tautological and Chow rings of Hurwitz moduli spaces leads to new results about the Chow rings of the moduli spaces of curves.

Ryan Contreras (Boston College): Plane $\mathbb{A}^1$-curves on the complement of strange rational curves

A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. We study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of $\mathbb{A}^1$-curves with a given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness of the complement using $\mathbb{A}^1$-curves parameterized by each irreducible component. I'm going to explain how the key to these results are the strangeness of all $\mathbb{A}^1$-curves.

Sarah Frei (Rice): Reduction of Brauer classes on K3 surfaces

For a very general polarized K3 surface over the rational numbers, it is a consequence of the Tate conjecture that the Picard rank jumps upon reduction modulo any prime. This jumping in the Picard rank is countered by a drop in the size of the Brauer group. In this talk, I will report on joint work with Brendan Hassett and Anthony Várilly-Alvarado, in which we consider the following: Given a non-trivial Brauer class on a very general polarized K3 surface over Q, how often does this class become trivial upon reduction modulo various primes? This has implications for the rationality of reductions of cubic fourfolds as well as reductions of twisted derived equivalent K3 surfaces.

Levi Heath (Colorado State): Quantum Serre duality for quasimaps

Let X be a smooth variety or orbifold and let Z be a complete intersection in X defined by a section of a vector bundle E over X. Originally proposed by Givental, quantum Serre duality refers to a precise relationship between the Gromov--Witten invariants of Z and those of the dual vector bundle E^\vee. In this talk, we present recent results proving a quantum Serre duality statement for quasimap invariants. In shifting focus to quasimaps, we obtain a comparison that is simpler and which also holds for non-convex complete intersections. This is joint work with Mark Shoemaker.

Kai Huang (MIT): K-stability of Log Fano Cone Singularities

We generalize the valuative criterion for K-stability of Fano varieties to log Fano cone singularities. We also show the higher rank finite generation conjecture for log Fano cone singularities, which implies the Yau-Tian-Donaldson Conjecture for Sasakian-Einstein metric.

Lena Ji (Michigan): The Noether–Lefschetz theorem in arbitrary characteristic

The classical Noether–Lefschetz theorem says that for a very general surface S of degree 4 in P^3 over the complex numbers, the restriction map from the divisor class group on P^3 to S is an isomorphism. In this talk, we will show a Noether–Lefschetz result for varieties over fields of arbitrary characteristic. The proof uses the relative Jacobian of a curve fibration, and it also works for singular varieties (for Weil divisors).

John Kopper (Penn State): Ample stable vector bundles on rational surfaces

A theorem of Fulton says that ample vector bundles cannot be classified numerically. However, ampleness is open in families, and so producing a single ample bundle typically implies the existence of many more. If a bundle is both stable and ample, then it has stable and ample deformations. Le Potier suggests exploiting this fact and classifying those Chern characters for which the general stable bundle is ample (provided, say, the moduli space is irreducible). I will discuss recent progress on this problem on the minimal rational surfaces. I will give a complete classification of those Chern characters for which the general stable bundle is both ample and globally generated. I will also explain an "asymptotic" version of this result for bundles that aren't globally generated. This is joint work with Jack Huizenga.

Joaquín Moraga (Princeton): Reductive quotient of klt singularities

In this talk, I will explain recent progress towards the understanding of quotients of smooth points by the action of reductive groups. The main result is that these quotients belong to the singularities of the minimal model program. Some applications of this result to moduli theory will be explained.

Jack Petok (Darthmouth): Kodaira dimensions of some moduli spaces of hyperkähler fourfolds

The Noether-Lefschetz locus in a moduli space of K3^[2]-fourfolds parametrizes fourfolds with Picard rank at least 2. Following Hassett’s work on cubic fourfolds, Debarre, Iliev, and Manivel showed that the Noether-Lefschetz locus in the moduli space of degree 2 K3^[2]-fourfolds is a countable union of special divisors indexed by discriminant d. In this talk, we compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show the divisors for discriminants greater than 224 are all of general type.

Nawaz Sultani (Michigan): Gromov--Witten invariants of some non-convex complete intersections

For convex complete intersections, the Gromov-Witten (GW) invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However, even in the genus 0 theory, the convexity condition often fails when the target is an orbifold, and so Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for orbifold complete intersections in stack quotients of the form [V // G]. This talk will be based on joint work with Felix Janda (Notre Dame) and Yang Zhou (Harvard), and upcoming work with Rachel Webb (Berkeley).

Fumiaki Suzuki (UCLA): An O-acyclic variety of even index

I will construct a family of Enriques surfaces parametrized by P^1 such that any multi-section has even degree over the base P^1. Over the function field of a complex curve, this gives the first example of an O-acyclic variety (H^i(X,O)=0 for i>0) whose index is not equal to one, and an affirmative answer to a question of Colliot-Thélène and Voisin. I will also discuss applications to related problems, including the integral Hodge conjecture and Murre’s question on universality of the Abel-Jacobi maps. This is joint work with John Christian Ottem.

Rachel Webb (Berkeley): The moduli of maps has a canonical obstruction theory

I will explain why the moduli of maps from tame twisted curves to a fairly general algebraic stack carries a canonical obstruction theory. A key ingredient is the construction of a dualizing sheaf and trace map for families of tame twisted curves.

Max Weinreich (Brown): The pentagram map

The pentagram map was introduced by Schwartz as a dynamical system on polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that, after a birational change of coordinates, it is a translation on a family of real tori. We will explain how the real, complex, and finite field dynamics of the pentagram map are all related by the following generalization: the pentagram map is birational to a translation on a family of Jacobian varieties.

Wern Yeong (Notre Dame): Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

A complex algebraic variety is said to be hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. Demailly introduced algebraic hyperbolicity as an algebraic version of this property, and it has since been well-studied as a means for understanding Kobayashi’s conjecture, which says that a generic hypersurface in dimensional projective space is hyperbolic whenever its degree is large enough. In this talk, we study the algebraic hyperbolicity of very general hypersurfaces of high bi-degrees in Pm x Pn and completely classify them by their bi-degrees, except for a few cases in P3 x P1. We present three techniques to do that, which build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As another application of these techniques, we simplify a proof of Voisin (1988) of the algebraic hyperbolicity of generic high-degree projective hypersurfaces.

Claudia Yun (Brown): Homology representations of compactified configurations on graphs

The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider each irreducible $S_n$-representation individually, vastly simplifying the calculation of these homology representations from the free resolution. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish.

Aline Zanardini (Leiden): The moduli space of rational elliptic surfaces of index two

Elliptic surfaces are ubiquitous in Mathematics. Examples include Enriques surfaces, Dolgachev surfaces, and all surfaces of Kodaira dimension one. In this talk we will focus on those elliptic surfaces which are rational and that have exactly one multiple fiber of multiplicity two. These are called rational elliptic surfaces of index two. Our goal will be to describe how to construct their moduli space when the choice of a bisection is part of the classification problem. This is based on work in progress joint with Rick Miranda.

The Boston College Organizers: Dawei Chen, Qile Chen, Maksym Fedorchuk, Brian Lehmann

We acknowledge the support of the NSF and the host institutions.