Algebraic Geometry Seminar

When we count rational points of bounded height on algebraic varieties, it is important to exclude exceptional sets to capture the generic distribution of rational points on underlying varieties. This idea leads to the notion of balanced line bundles, and the study of balanced line bundles can be achieved through the study of birational geometry, e.g., the Minimal Model Program. In this talk, first I introduce height functions and counting problems, and I discuss the notion of balanced line bundles. Then I will talk about how useful birational geometry is to study this notion.

The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series of dimension one, the Castelnuovo specialize to the better known Catalan numbers. In this talk, I will present a formula for the number of linear series on a general curve with prescribed ramification at an arbitrary point, when the expected number is finite. As an application, I will show how to solve certain enumerative problems on moduli spaces of curves, and how to obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a general curve. This is a joint work with Gavril Farkas.

Log canonical thresholds are important invariants of singularites of pairs which play an important role in higher dimensional birational geometry (and many other subjects). Shokurov conjectured that (in a fixed dimension) the set of all possible log canonical thresholds T_n should enjoy some remarkable properties. In this talk we will discuss the proof of Shokurov's ACC for LCTs conjecture stating that if t_i\in T_N is a non-decreasing sequence, then t_i is eventually constant.

In the early days of mirror symmetry, physicists noticed a remarkable relation between the Calabi-Yau geometry of a hypersurface in projective space defined by a homogenous polynomial W and the singularity theory of the Landau-Ginzburg model with superpotential W. This relation came to be known as the Landau-Ginzburg/Calabi-Yau correspondence. In this talk, I will explain how this correspondence can be extended to non-Calabi-Yau hypersurfaces in weighted projective space using the recently introduced Fan-Jarvis-Ruan-Witten theory as the mathematical formalism behind Landau-Ginzburg models.

In this talk, I will explain how to compute FJRW invariants in genus zero for a large class of polynomials which do not satisfy the concavity hypothesis. This should be seen as the counterpart of Gromov--Witten theory for hypersurfaces where convexity fails. Moreover, to get a complete description of the FJRW quantum theory for these polynomials, we will also have to deal with matrix factorizations, using an algebraic construction of Polishchuk and Vaintrob. At last, we end up with a new characteristic class in K-theory, leading to the expression of the virtual class and to a mirror statement. Interestingly, we also obtain with the same ideas some FJRW invariants in arbitrary genus.

We prove Weak Borisov-Alexeev-Borisov Conjecture in dimension three which states that the anti-canonical volume of an $\epsilon$-klt log Fano pair of dimension three is bounded from above.

The goal of this seminar is to review and contextualize the evolution of my thoughts and interactions with tropical geometry. This talk is based on collaborative work always with Hannah Markwig, and at different times with each one of Aaron Bertram, Paul Johnson and Dhruv Ranganathan. Back in 2007, Hannah Markwig approached me after being told by Paul Johnson that her tropical covers "smelled" like cut and join. Deciphering Paul’s oracle was the beginning of a fruitful and ongoing collaboration, that is pulling me closer and closer to the tropical world. Over the course of the years, we have been studying Hurwitz theory and Gromov-Witten theory, first using tropical geometry as a powerful combinatorial tool, and then trying to understand what is the conceptual reason for the remarkably tight connection between the boundary geometry of moduli spaces of curves and maps and the piecewise linear objects in tropical geometry. The introduction of the analityc point of view, brought to the moduli space of curves by Abramovich, Caporaso and Payne, offered not only a much sought for conceptual perspective, but also opened up the way for further investigation.

Let M be a moduli scheme of stable sheaves on a complex elliptic surface. We want to know its birational structure, especially its Kodaira dimension. For this end, it is important to understand its singularities. What is known about such problems now?

One of the fundamental invariants for families of curves is the slope. In this talk, I will state a slope inequality for families of curves over surfaces. In fact, it is closely related to the "slope" in the geography of irregular varieties. I will also introduce this notion and present some recent results for irregular 3-folds of general type.

In “Equations of tropical varieties,” J.H.Giansiracusa and I introduced a scheme-theoretic framework for tropicalization and tropical geometry. In this talk I’ll discuss recent developments in this program. Specifically, we introduce a canonical embedding of any scheme in an F1-scheme (in essence, a non-finite type toric variety) such that the corresponding tropicalization is the inverse limit of all tropicalizations and its T-points form the space underlying Berkovich’s analytification. This is related to Payne’s topological inverse limit result.

Let X be a smooth canonically polarised surface defined over an algebraically closed field of characteristic p>0. In this talk I will present some results about the geometry of X in the case when the automorphism scheme Aut(X) of X is not smooth, or equivalently X has nontrivial global vector fields. This is a situation that appears only in positive characteristic and is intimately related to the structure of the moduli stack of canonically polarised surfaces in positive characteristic because the smoothness of the automorphism scheme is the obstruction for the moduli stack to be Deligne-Mumford, something that is always true in characteristic zero but not in general in positive characteristic. One of the results that will be presented in this talk is that smooth canonically polarised surfaces with non smooth automorphism scheme and “small” invariants are algebraically simply connected and uniruled.

For a fixed Calabi-Yau threefold X, Donaldson-Thomas (DT) theory, roughly, is the study of certain Euler characteristics of Hilbert schemes of curves in X. If X is an orbifold with crepant resolution Y, Bryan, Cadman, and Young conjectured that the DT theory of X and Y should be related in a simple way. We prove this conjecture in the toric setting. In this talk, I'll begin by describing the basic notions of DT theory and motivate them through the concrete example of toric varieties. I'll explain how these notions generalize to orbifolds and describe some of the techniques used in the proof of the correspondence.

Abstract: In many instances one can define the notion of volume of a representation of the fundamental group of a closed manifold M into a simple (non-compact) Lie group G. This is so for instance if M is a surface and the symmetric space associated to G is hermitian, that is carries an invariant 2-form, or if M is a 3-manifold and G is a complex group, equivalently the associated symmetric space carries an invariant 3-form. When M is not compact the definition of volume of a representation presents interesting difficulties; in this talk we will show how bounded cohomology can be used to define an invariant generalizing the volume of a representation and we will see how this invariant is connected with the deformation theory of such representations. This is joint work with Michelle Bucher and Alessandra Iozzi.

Stable varieties are higher dimensional generalizations of stable curves. Their moduli space contains an open locus parametrizing varieties of general type up to birational equivalence, just as the space of stable curves contains the space of smooth curves in dimension one. Furthermore, also similarly to the one dimensional picture, it provides a compactification of the above locus, which is known in characteristic zero but it is only conjectural in positive characteristic in dimension at least two. I will present a work in progress aiming to prove the projectivity of every proper subspace of the moduli space of stable surfaces in characteristic greater than 5.

Quantum curves are conceived in physics as a result of geometric quantization of the SL(2) character variety of a knot group, which is an infinite-order differential operator that characterizes the colored Jones polynomial. The physics speculation relates the character variety, which is an algebraic curve defined over the integer ring, and the colored Jones polynomial through the topological recursion of Eynard, Orantin, and their collaborators.

A mathematical relation between the topological recursion and quantum curves has been discovered in my joint work with Olivia Dumitrescu in the context of Hitchin's theory of Higgs bundles on a curve. Although it has nothing to do with knot polynomials, the geometric structure becomes clear i this theory. This talk surveys recent mathematical developments around quantum curves and topological recursion, both concrete examples and an algebraic geometry theory of the notion.