Minicourses
Title: Cohomology of the moduli space of curves.
Abstract: I will explain several new ideas leading to a better understanding of the cohomology of moduli spaces of (stable) curves of all genera. Joint with Hannah Larson, Sam Payne, and Thomas Willwacher.
Title: Geometric and tropical aspects of compactified Jacobians of nodal curves
Abstract: The study of Jacobians and their compactifications is central to understanding the geometry of algebraic curves, especially in the context of degenerations and moduli. Compactified Jacobians provide essential tools for extending classical theorems and constructions to singular curves, revealing deep connections between algebraic geometry, combinatorics, and tropical geometry. This mini-course will explore these rich interactions. We will begin with the theory of compactified Jacobians for nodal curves. We will then introduce tropical Jacobians, showing how they arise combinatorially from metric graphs. A central focus will be the Abel map for nodal curves and its resolution, highlighting the interplay between algebraic and tropical geometry. Finally, we will discuss the Torelli theorem for nodal curves, addressing the question of reconstructing a curve from its compactified Jacobian. The course aims to provide a bridge between the classical geometric theory of Jacobians and the modern perspectives offered by tropical geometry and degeneration techniques.
Title: Tropical geometry and applications
Abstract: The objective of this course is to give an introduction to the main ideas of tropical geometry and some of its applications. My plan is to cover the following four blocks (roughly fitting the four classes of the course): Tropical varieties and tropicalization, tropical intersection and Hodge theory, tropical patchworking of real varieties, tropical enumerative geometry.
Title: Brill-Noether theory via degeneration
Abstract: Brill--Noether theory is the representation theory of abstract curves, asking: given an abstract curve, what are all of the ways that I can represent it as a curve with some invariants in projective space? The famous Brill--Noether theorem answers this when the abstract curve is general in the moduli space of genus g curves. It is difficult to get your hands on a ''general curve'', but specialization to degenerate curves can shed light on the behavior for general curves. In the first three lectures of this minicourse, we will study limits of line bundles under degeneration to prove some aspects of the Brill--Noether theorem. In the final lecture, we will turn our attention to curves that are not general by virtue of having an unexpected low degree map to P^1, and see how these ideas can also be adapted to this situation.
Talks
Title: Irreducibility of Severi varieties on toric surfaces
Abstract: Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will present some results on the irreducibility of these varieties in the case of toric surfaces, and their application to the irreducibility of other moduli spaces of curves. This is done using tropical methods and I will indicate some of these aspects. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.
Title: Computing the base change conductor for Jacobians
Abstract: Let K be a discretely valued field, with ring of integers R.The base change conductor of an abelian K-variety, denoted c(A), is a numerical invariant which measures the failure of A to have semi-abelian reduction over R. It can be difficult in general to compute c(A) explicitly. In this talk I will present an approach for Jacobians, using intersection theory and invariants of quotient singularities on certain normal R-models of the curve in question. (Joint work with D. Eriksson and J. Nicaise).
Title: Compact degenerations of curves and derived categories
Abstract: Consider a compact degeneration of curves, i.e., a flat family of curves over a disc such that the central fiber is a 1-nodal curve with two irreducible components and all other fibers are smooth. I will explain how the family of derived categories of the smooth fibers is related to a category glued from the derived categories of the components of the central fiber. This is a joint work in progress with Valery Alexeev.
Title: Ranks for divisors on graphs and Clifford’s inquality for nodal curves
Abstract: Baker’s specialization lemma is a powerful technique to obtain results holding for general algebraic linear series on curves from considering the Baker-Norine rank of their their tropicalization. However, in the years there have been different proposals for notions of rank for graphs and tropical curves, which can be applied to different problems.
One such problem is the existence of special multidegrees of line bundles on reducible curves for which Clifford’s inequality still holds.
Indeed, Clifford’s inequality gives an upper bound for the rank of a line bundle of given degree on a smooth curve of given genus, which remains valid for irreducible curves, but could never hold in general for curves with several components.
In the talk, I will discuss joint work with M. Barbosa and K. Christ where we use a variant of the so-called algebraic rank to establish the existence of such special multidegrees.
Title: Fourier transforms and Abel-Jacobi theory
Abstract: One of the central objects in the intersection theory of curves is the tautological ring. In the last 30 years, remarkable progress has been made in our understanding of the tautological ring, largely due to the connection between the geometry of curves and the geometry of stable maps. Compactified abelian fibrations, and compactified Jacobians in particular, also have tautological rings, but the study of their structure requires different ideas. In this talk, I will explain how the structure of these rings can be controlled through the interaction between certain Fourier-Mukai transforms and logarithmic geometry. This is based on joint work with Bae and Pixton.
Title: On the Poincaré polynomial of the moduli space M0,n
Abstract: We present old and novel formulas for the Betti numbers of the compactified moduli space M0,n of genus-zero curves with n marked points. We provide a unified setting and new proofs of the famous results by Keel, Getzler, Manin, and Aluffi - Marcolli - Nascimento, clarifying the geometric intuition behind these formulas. Our techniques are based on the theory of lattices with building sets and classical results on the exponential generating functions. This is a joint work with Eur, Ferroni, Matherne, and Vecchi.
Title: Singular points on moduli of Fano varieties via toric geometry
Abstract: Thanks to K-stability (existence of Kähler-Einstein metrics), moduli spaces/stacks of Fano varieties have been constructed. In this talk I will explain how to use toric geometry and the combinatorics of lattice polytopes to produce examples of singular points on these spaces/stacks of Fano varieties. This talk is partly based on joint work with Anne-Sophie Kaloghiros.
Title: Vector bundles in tropical geometry: An elementary approach
Abstract: Tropical geometry studies a piecewise linear combinatorial shadow of degenerations and compactifications of algebraic varieties. A typical phenomenon is that many of the usual algebro-geometric objects have a tropical analogue that is intimately tied to its classical counterpart. An example is the theory of divisors and line bundles on
algebraic curves, whose tropical counterparts have been crucial in numerous surprising applications to classical Brill—Noether theory and the birational geometry of moduli spaces. One classical object that has resisted the effort of tropical geometers so far is the geometry of vector bundles beyond rank one. In this talk, I will outline an elementary approach to tropical vector bundles that builds on earlier work of Allermann. Although limited in scope, this theory leads to a satisfying tropical story for semistable vector bundles on elliptic curves and, more generally, semihomogeneous vector bundles on abelian varieties. The engines in the background that make these cases accessible to our methods are Atiyah's classification of vector bundles on elliptic curves, Fourier-Mukai transforms on abelian varieties, and the interactions with non-Archimedean uniformization. This talk is based on joint work with Andreas Gross and Dmitry Zakharov (and, in parts, Arne Kuhrs) as well as with Andreas Gross, Inder Kaur, and Annette Werner.
Lightning talks
Title: Counting tropical covers
Title: Using Graph Complexes to Study the Cohomology of the Moduli Space of Curves
Title: Vanishing H^1 of the Hurwitz space of admissible covers of low-degree
Abstract: Harris and Mumford introduced the ``admissible covers'' compactification of the moduli space of simply-branched covers, in which the target and source curves of an admissible cover degenerate into nodal curves as branch points come together. The boundary of this moduli space is stratified by Hurwitz spaces parametrizing more general covers. In this talk, we will describe the stratification of the boundary and how we can adapt an inductive method of Arbarello and Cornalba towards computing the vanishing H^1 of this moduli space for covers of degree <= 4.
Title: Tropical toric vector bundles
Abstract: Tropical geometry is a new branch at the interface between algebraic geometry and combinatorics. Tropical geometry is intimately connected with toric geometry. For instance, the tropicalization of a projective toric variety is homeomorphic to the associated lattice polytope. We introduce the notion of tropical toric vector bundles on tropical toric varieties using valuated matroids. Our motivation comes from the work of Di Rocco, Jabbusch, and Smith, where the authors associated realizable matroids to a toric vector bundle. We study splitability on tropical projective lines. In contrast to the classical situation, we show that tropical toric bundles on tropical projective line do not necessarily split. We also study the semistability of such bundles. This is a joint work with Diane Maclagan.
Title: The tropical Abel-Prym map and the bigonal construction
Title: The Prym-canonical Clifford index
Title: Exploring tropical linear series on tropical curves
Title: Logarithmic Geometry and the Operad of Little Disks
Abstract: The operads of little disks and framed little disks are arguably the most well known and important examples of topological operads. Due to reasons this talk is too short to mention, the dimension 2 versions of these operads have long been expected to have some form of algebro-geometric or motivic origin. This expectation was shown to be true by Dmitry Vaintrob who constructed a log-geometric model of the framed little 2-dimensional disks operad in 2019. In this talk I will review the main ideas of Vaintrob's construction and explain how they can be generalized to give a similar result for little disks in arbitrary even dimension using the moduli spaces of stable pointed trees of projective spaces.
Title: Parabolic sheaves in logarithmic geometry
Title: Syzygies of line bundles on Ribbons over curves
Title: Universal relations on moduli of twisted maps
Title: The Kodaira dimension of the moduli space of marked curves in genus 3
Abstract: For relatively low genus g the moduli space M_g is uniruled and therefore has negative Kodaira dimension.
When g grows, the birational geometry of M_g becomes more complicated and for g at least 22 the space M_g has maximal Kodaira dimension.
More generally, for g at least 2, the Kodaira dimension of the moduli spaces of pointed curves M_g,n is maximal in all but finitely many cases. For genus g=3, this result is new.
In this talk I will explain the main ideas behind the proof of the result in genus 3.
Title: Euler characteristics of stable curves and stable maps
Title: Subvarieties in Abelian Variety
Abstract: Abelian varieties contain many interesting subvarieties that reflect their rich geometry. In this talk I will describe a construction of families of such subvarieties starting from a fixed subvariety. I will describe their dimensions and homology classes in specific cases, with explicit results in the Jacobian case.