Boston College Algebraic Geometry Seminar

Abstract: I will explain the existence of a non-tautological class on the moduli space of principally polarized abelian sixfolds. The proof uses the pullback to the moduli space of genus 6 curves of compact type via the Torelli morphism. In a surprising twist, the proof sheds light on Pixton’s conjecture for the structure of the tautological ring of moduli spaces of curves. This is joint work with Dragos Oprea and Rahul Pandharipande.

We investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras.  We construct a trace map on the elliptic chiral homology of the chiral \beta-\gamma-bc system and explain a chiral analogue of Hochschild-Kostant-Rosenberg theorem. A geometric renormalization theory for 2d chiral QFT is established. leading to a version of holomorphic anomaly equation.

Abstract: Solomon Lefschetz showed that the Picard group of a general surface in P3 of degree greater than three is ZZ. That is, the vast majority of surfaces in P3 have the smallest possible Picard group. The set of surfaces of degree greater than 3 on which this theorem fails is called the Noether-Lefschetz locus. This locus has infinite components and their dimensions are somehow mysterious.  In this talk, I will describe infinitely many Noether-Lefschetz components and compute their dimension. This will exhibit infinitely many components that were previously unknown. This is joint work with Montserrat Vite and Manuel Leal.

The finite Hecke category of Borel-constructible sheaves on the flag variety admits an interesting auto-equivalence that exchanges projectives for simples. One consequence is Soergel’s endomorphismensatz: the projective cover of the skyscraper sheaf has endomorphisms equal to the cohomology of the flag variety. I will discuss an 'uncompletion' where cohomology is replaced by equivariant K-theory. This is a first step towards tamely ramified local Betti geometric Langlands (uncompleting Bezrukavnikov’s equivalence).

Abstract: This talk will be about the cohomology ring of moduli spaces (and moduli stacks) of semistable 1-dimensional sheaves on P^2. I will explain an approach to describing these rings in terms of generators and relations which allowed us to completely determine the cohomology rings of moduli spaces of sheaves supported in curves up to degree 5. I will discuss the perverse filtration in the cohomology of such spaces, its relation to curve counting invariants and a new conjectural description of it. This is joint work with Yakov Kononov, Woonam Lim and Weite Pi.

Abstract: The tautological ring is a subring of the cohomology of the moduli space of stable curves, which is simultaneously tractable to study and yet rich enough to contain most cohomology classes of geometric interest.  The first known example of a non-tautological algebraic class was discovered by Graber and Pandharipande, in work that was later significantly generalized by van Zelm to produce an infinite family of non-tautological classes on the moduli space of stable curves.  On the other hand, very little was previously known about non-tautological classes on the moduli space of smooth curves: for only finitely many genera and numbers of marked points were such classes known to exist.  I will report on joint work with V. Arena, S. Canning, R. Haburcak, A. Li, S.C. Mok, and C. Tamborini (from the 2023 AGNES Summer School), in which we produce non-tautological algebraic classes on the moduli space of smooth curves in an infinite family of cases, including on M_g for all g>15.

Abstract: In recent years, the intersection theory of tropical fans has formed a bridge between algebraic geometry and combinatorics. In this talk, I’ll describe this bridge and discuss a method for importing tools from convex geometry into the study of intersection theory on tropical fans.

Abstract: Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will discuss the classical question of whether Severi varieties are irreducible and its relation to the irreducibility of other moduli spaces of curves. I will try to indicate how new, tropical methods can be used to answer such irreducibility questions. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.

Abstract: I'll present ideas of an ongoing project, joint with Alvarez-Gavela and Eliashberg, to construct geometric spaces from their homological invariants. Specifically, we would like to construct Lagrangian submanifolds filling Legendrian submanifolds from sheaves with prescribed microlocalizations.  

Abstract: Recently, we developed an approach to the K-moduli of Fano threefolds, based on the moduli continuity method. We apply it to the family of blow-ups of 3-dimensional projective space along (2,3)-complete intersection curves, and describe the K-moduli space explicitly. The two main ingredients of this approach are the moduli of K3 surfaces, and general elephants.

Abstract: The complexity is an invariant introduced by Shokurov in order to understand whether a log Calabi-Yau pair is toric.  In this talk, I will introduce a birational variant of the complexity, the so-called birational complexity. I will explain how it can be used to detect whether a Fano variety X admits a boundary B such that (X,B) can be birationally transformed into a toric log Calabi-Yau pair. Parts of this talk are joint work with Mirko Mauri, Fernando Figueroa, Joshua Enwright, Konstantin Loginov, and Artem Vasilkov.

Abstract: In this talk we will be concerned with smooth, framed fiber bundles whose fibers are the standard d-dimensional disk, trivialized along the boundary. "Kontsevich's characteristic classes" are invariants defined for these bundles: given such a bundle \pi:E \to B, we can associate to it a collection of cohomology classes in H^*(B). On the other hand, there is a "bracket operation" for these bundles defined by Sander Kupers: namely, given two such bundles \pi_1 and \pi_2 as input, we can output a "bracket bundle" [\pi_1,\pi_2]. I will talk about this bracket bundle construction and a formula relating the Kontsevich's classes of [\pi_1,\pi_2] with those of \pi_1 and \pi_2. The main input of the proof is a novel but very natural configuration space generalizing the Fulton-MacPherson configuration spaces. This is a work in progress joint with Robin Koytcheff and Sander Kupers.