Boston College Algebraic Geometry Seminar

Abstract: Just as Heegaard-Floer (HF) theory is about Fukaya categories for symmetric product of Riemann surfaces, HF theory with coefficient is about  Fukaya categories of horizontal Hilbert scheme (possibly with fiberwise superpotential). This is used to give a symplectic realization of categorified quantum group representation. The method of cutting and gluing Riemann surface can be used to take tensor product of categorified representations. I will give the example when g=sl_2. This is work in progress, partly joint with Vivek Shende, and partly joint with Mina Aganagic, Elise LePage, Yixuan Li. 

When the invariant ring of a reductive group action on a variety is isomorphic to the invariant ring of a finite group action on a closed subvariety, we say that this subvariety has the Chevalley restriction property. The well-known Chevalley restriction theorem says that the Cartan subalgebra of a semisimple Lie algebra has this property, and results of Dadok-Kac and Luna-Richardson identify analogous subspaces for some representations of reductive groups. We will discuss a work in progress which further extends this story to the level of Galois theory and show some connections to period integrals of Calabi-Yau families. This is joint work with Bong Lian.

Given a smooth curve C, it is natural to ask: what are all the degree d maps from C into a projective space $\mathbb{P}^r$? The study of this question is called Brill-Noether theory. Given a curve $C$, the data of a degree d map $C \rightarrow \mathbb{P}^r$ is equivalent to the data of a degree d line bundle on $C$ together with a choice of $r + 1$ global sections having no common zeros. As such, a central object of study is the Brill–Noether locus $W^r_d(C)$, which is defined to be the space of degree $d$ line bundles on $C$ with at least $r+1$ global sections.

The famous Brill-Noether theorem gives a nice description of $W^r_d(C)$ when C is a general curve of genus g. However, curves we come across in nature (such as curves in the plane) are not general, and may fail the Brill-Noether theorem!  In this talk, I'll describe joint work with Hannah Larson, in which we describe the Brill-Noether theory of smooth plane curves (and more generally, curves on Hirzebruch surfaces), using tools from arithmetic statistics. 

Reider's theorem gives a criterion for |K_X+D| to be globally generated or very ample when D is a nef divisor on a smooth projective complex surface. Reider's original proof used Bogomolov's work on stability, but later authors have reproved the result using techniques ranging from vanishing theorems (Sakai) to Bridgeland stability (Arcara-Bertram). In joint work with Anda Tenie, we use this latter approach, combined with Langer's recent construction of Bridgeland stability conditions on normal surfaces in arbitrary characteristic, to prove Reider-type results on separation of zero-dimensional subschemes on normal surfaces.

Rational curves play a critical role in understanding the birational geometry of varieties. Free curves are the easiest to work with, but on Fano varieties that are even mildly singular, it remains an open question whether these free rational curves exist. In this talk, we discuss free curves of higher genus. Using some ideas on stability of vector bundles, we show that any klt Fano variety has these higher-genus free curves. We then use the existence of these free curves to get some applications, including the existence of free rational curves in terminal Fano threefolds, the lengths of extremal rays of the cone of curves, and studying the fundamental group of the smooth locus of a terminal variety. This is joint work with Eric Jovinelly and Brian Lehmann.

The topology of algebraic function spaces has been a topic of great interest for decades, as exemplified by the Cohen-Jones-Segal conjecture. Moreover, through the Weil conjectures, the topology of certain moduli spaces has had a profound impact on questions in number theory—one striking example being the work of Ellenberg, Venkatesh, and Westerland on the Cohen-Lenstra heuristics. In this talk, I will outline a framework for studying the cohomology of the moduli space of algebraic maps by leveraging well-structured hypercovers. In particular, our methods confirm the geometric Manin’s conjecture for complete simplicial toric varieties over global function fields.

Abstract:  Let J_G be the universal regular centralizer group scheme for a reductive group G.  To a tempered affine spherical G-variety X, we associate a subgroup of J_G, which is Lagrangian in J_G, and a polynomial depending on the geometry of the action of G on T^*X. Following the philosophy of relative Langlands as developed by Ben-Zvi, Sakellaridis, and Venkatesh, we give conjectural interpretations of these invariants via data associated to their conjectural dual Hamiltonian. We will give a global application to the relative duality of Hitchin systems, which generalizes a conjecture of Hitchin.

The Picard rank conjecture predicts the vanishing of the rational picard group of the Hurwitz space parameterizing simply branched covers of $\mathbb P^1$ of degree $d$ and genus $g$.

In joint work with Ishan Levy, we prove the Picard rank conjecture when $g$ is sufficiently large relative to $d$. The main input is a new result in topology where we prove that the homology of Hurwitz spaces stabilize and compute their stable value.

In 1995 Mukai showed that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian OG(5,10) in P^15 with a six-dimensional projective linear subspace, and that the GIT quotient Gr(7,16)//Spin(10) is a birational model of the moduli space of curves $\bar{M}_7$. Which singular objects appear on the boundary of Mukai's model? As a first step in this study, calculations in Macaulay2 and Magma are used to find and analyze linear spaces yielding three singular curves: a 7-cuspidal curve, the balanced ribbon of genus 7, and a family of genus 7 graph curves.

The category of perverse sheaves is defined as an abelian subcategory of the derived category of constructible sheaves. This notion has a coherent counterpart, defined inside the derived category of coherent sheaves. Cautis and Williams conjectured that the category of perverse coherent sheaves on the affine Grassmannian is a cluster monoidal categorification. We discuss a partial progress in this conjecture. Our approach is based on relating this category to the category of modules over quantum loop group.

Given a smooth projective curve X, there are several naturally defined vector bundles on X such as the normal bundle of X or the restriction of the tangent bundle of projective space to X. Similarly given a branched covering of curves $f: Y \to X$, there are several natural vector bundles such as the Tschirnhausen bundle on X associated with the covering. In this talk, I will discuss the stability of normal bundles of Brill-Noether curves and Tschirnhausen bundles. The talk is based on joint work with Eric Larson, Isabel Vogt, Geoffrey Smith and Eric Jovinelly.

I will discuss the idea of proof of birational rigidity of threefolds and the importance of local inequalities for their proof. Then I will discuss the birational rigidity results that follow from local inequalities for cA_n points, in particular I will talk about birational rigidity of sextic double solids with cA_n-singularities. At the end of the talk I will talk a bit about the differences in approach that allow us to get inequalities extended from cA_1-points to cA_n-points.