I will continue a long struggle to explain to non-experts why resolution of singularities in characteristic zero works. This is joint work with Michael Temkin and Jaroslaw Wlodarczyk.

For every family of smooth curves degenerating to a fixed stable curve on the Deligne-Mumford boundary of the moduli space of curves, the canonical linear series of Abelian differentials degenerate to a collection of linear series on the limiting singular curve. As the degenerating family varies, so does the resulting collection. In this way, each stable curve in the moduli space is associated with infinitely many such collections of linear series - one for each degenerating family. We describe all limit collections of linear series for stable curves that are general within a given dual graph stratum, in the sense that the markings given by the nodes on each irreducible component of the stable curve are in general position. We also construct a projective variety which parametrizes these collections. This had previously been achieved only in very special cases: for curves of compact type by Eisenbud and Harris in the 1980s, for two-component curves by Esteves and Medeiros in the 2000s, and in a few other isolated situations. The breakthrough was made possible by combining work by Bainbridge, Chen, Gendron, Grushevsky and Möller from 2016 on the residue conditions limits of differentials must satisfy, with recent work by ourselves on the tropicalization of general linear series. We will review the history - specially the latter work - and explain our analysis and construction. Time allowing, we will also discuss related projects on the subject.

Let \(C\) be a curve of genus \(>2\) , embedded in its Jacobian \(J_C\). The cycle \([C] - [(-1)^* C]\) is cohomologous to zero in \(J_C\); is it algebraically equivalent to zero? The answer is negative for \(C\) general (Ceresa, 1983), and positive (trivially) for hyperelliptic curves. In the last three years a number of approaches have been developed to find non-hyperelliptic curves for which this cycle is algebraically trivial. I will survey the history of the problem, then discuss these recent examples of non-hyperelliptic curves, in particular the approach of Laga and Shnidman (2024).

Let \(X\) be a smooth and complex Fano 4-fold, and \(\rho(X)\) its Picard number. We will discuss the following theorem (in progress): if \(\rho(X)>9\), then \(X\) is a product of del Pezzo surfaces. Note that this is sharp, since there is a family of Fano 4-folds with \(\rho(X)=9\) that is not a product of surfaces. We will give an overview of the proof, based on a detailed and explicit study of the geometry of Fano 4-folds with \(\rho(X)>6\), using birational geometry in the framework of MMP. A key ingredient are the properties of the Lefschetz defect, an invariant that relates the Picard number of \(X\) to that of its prime divisors.

My research has been very strongly influenced by Lucia's work over the past decade on curves, Jacobians, and moduli spaces in tropical geometry. In this talk, I will discuss recent joint work with Eran Assaf, Madeline Brandt, Juliette Bruce, and Raluca Vlad, in which we develop the theory of tropicalizations of locally symmetric varieties and study its consequences for cohomology of moduli spaces and cohomology of arithmetic groups.

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.

Let \(X\) and \(Y\) be subvarieties of a simple abelian variety \(A\) such that \(Z=X+Y\ne A\). We prove that the sum map \(X\times Y \to Z\) is semismall. In particular, \(Z\) has the expected dimension \(\dim(X)+\dim(Y)\). Over the field of complex numbers, the latter statement was proved in 1982 by Barth, and Prasad gave in 1993 a very simple proof. Surprisingly enough, it is not known in general in positive characteristics. I will discuss some cases where these statements hold. This is joint work with Ben Moonen.

For every family of smooth curves degenerating to a fixed stable curve on the Deligne-Mumford boundary of the moduli space of curves, the canonical linear series of Abelian differentials degenerate to a collection of linear series on the limiting singular curve. As the degenerating family varies, so does the resulting collection. In this way, each stable curve in the moduli space is associated with infinitely many such collections of linear series - one for each degenerating family. We describe all limit collections of linear series for stable curves that are general within a given dual graph stratum, in the sense that the markings given by the nodes on each irreducible component of the stable curve are in general position. We also construct a projective variety which parametrizes these collections. This had previously been achieved only in very special cases: for curves of compact type by Eisenbud and Harris in the 1980s, for two-component curves by Esteves and Medeiros in the 2000s, and in a few other isolated situations. The breakthrough was made possible by combining work by Bainbridge, Chen, Gendron, Grushevsky and Möller from 2016 on the residue conditions limits of differentials must satisfy, with recent work by ourselves on the tropicalization of general linear series. We will review the history - specially the latter work - and explain our analysis and construction. Time allowing, we will also discuss related projects on the subject.

We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line, via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz-Brill-Noether theory. In particular, we provide new proofs of the main non-existence and existence results in Hurwitz-Brill-Noether theory. Finally, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz-Brill-Noether theory. Joint work with Soheyla Feyzbakhsh and Andres Rojas.

Some 40 years ago, Faltings proved the remarkable theorem that a curve of geometric genus \(g > 1\) over a number field \(K\) can have only finitely many rational points over \(K\). This immediately gave rise to two follow-up questions. First, we can ask if the number is bounded in terms of \(g\) and \(K\). And second, we can ask if there is an analogue of Faltings' theorem for higher-dimensional varieties. In this talk, I'll describe what is conjectured about these questions, as well as an unexpected connection between the two.

This talk explores the rich interplay between line bundles on algebraic curves and chip-firing games on metric graphs. I will begin by revisiting earlier joint work with Lucia Caporaso and Margarida Melo on the relationship between the algebraic and tropical ranks of divisors, highlighting how this framework helped bridge the gap between the classical and tropical worlds. I will also outline subsequent developments on this problem. In the second part of the talk, I will present recent joint work with Thibault Poiret, in which we use tropical techniques to construct a universal space for Prym varieties. This project extends Caporaso's influential work on compactified Jacobians and provides new tools for studying degenerations of abelian varieties in families.

TBA

Given a line bundle L over the moduli space of curves, the space parametrizing r-th roots of L yields a natural finite cover of the space of curves. Spaces of roots are very interesting as they carry lots of geometrical information on the spaces of curves themselves and can be very conveniently applied in enumerative problems. Starting from the well known cases of spin and r-th spin curves, we will discuss how to study stratifications of nice compactifications of these spaces over the moduli space of stable curves using logarithmic geometry.

The classical Torelli theorem states that a smooth curve can be recovered from its polarized Jacobian. In this talk, we explore extensions of this theorem to stable curves and their dual graphs. A central result in this context is the work of Caporaso and Viviani, which establishes a Torelli theorem for stable curves via coarse compactified Jacobians. We will discuss the dependence of this result on the concept of compactified Jacobians and present an alternative approach to their proof. This is a joint work in progress with Alex Abreu and Nicola Pagani.

I will explain recent directions in the study of the geometry and cycle theory of the moduli space of compactified Jacobians. Along the way, I will discuss the work of many mathematicians over the years starting, of course, with Caporaso's PhD thesis in 1993.

I will survey recent advances in understanding unstable cohomology groups of moduli spaces of curves using algebraic geometry. The proofs involve an intricate interplay between geometric topology, Hodge theory, and point counting over finite fields, governed by the combinatorial and inductive structure of the boundary of the stable curves compactification. In particular, this work builds on the understanding developed a decade ago in joint work with Abramovich and Caporaso on the tropicalization of the moduli space of curves.

We study normal functions (sections of the Jacobian bundle) defined on the moduli space of pointed curves. Using the infinitesimal Griffiths invariant (refined by M. Green and C.Voisin) we show that a normal function with nontrivial but sufficiently "small" support cannot be "locally constant". As an application, we give some results for very general curves and very general plane curves. This is a joint work with Lorenzo Fassina.

In the 1990's A. Treibich and J.L. Verdier studied a class of curves of genus \(g\), for any \(g \geq 3\), which enjoy some remarkable properties. In particular Treibich proved that they are Brill-Noether general. I will survey the main properties of such curves and outline Treibich's proof.

We will discuss the three incarnation of the exceptional set appearing in conjectures of Lang, Vojta, Green and Griffiths, focusing on the case of projective space. In particular we will present recent descriptions of the algebraic exceptional set obtained for complements of curves in the projective plane and some generalizations to other surfaces. (based on joint works with Lucia Caporaso).

I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.

If \(G\) is a finite group, its Cremona dimension is the least \(n\) such that \(G\) is a subgroup of the group of birational automorphisms \(Bir(X)\) of a rationally connected projective variety \(X\). Until 2009 we had a good understanding of the groups of Cremona dimension at most 2, that is, the finite subgroups of the classical Cremona group \(Bir(\mathbb{P}^2)\), but no examples where known of a finite group of Cremona dimension larger than 3. Since then the progress has been rapid. I will review the work that has been done on this, particularly by Prokhorov and Shramov, and state a remarkable fixed point theorem due to Haution, which in particular implies that if \(p\) is a prime, a non-abelian p-group has Cremona dimension at least \(p-1\). Then I will explain an improvement of this last result, due to G. Bresciani, Z. Reichstein and myself, which gives more refined lower bounds for the Cremona dimension of a p-group. This relies on a new technique, connecting Cremona dimension with relative Brauer groups for projective varieties over non-algebraically closed fields.

We study the problem of classifying compactified Jacobians of nodal curves that can arise as limits of Jacobians of smooth curves. The answer is given in terms of a new class of compactified Jacobians, that is strictly larger than the class of classical compactified Jacobians, as constructed by Oda-Seshadri, Simpson, Caporaso and Esteves. A consequence of our result is a complete classification of all the modular compactifications of the universal Jacobian over the moduli stack of pointed stable curves. This is based on a joint work with M. Fava and N. Pagani.

The Abel-Jacobi map on cycles algebraically equivalent to zero generalizes the Albanese map on zero-cycles. For divisors, the Abel-jacobi map provides an isomorphism between the subgroup of the Picard group made of divisors homologous to zero and the first intermediate Jacobian, and it is known to have wonderful properties; in particular there exists a Poincaré divisor, or universal divisor parameterized by the Jacobian. I will discuss the analogous geometry for higher codimension cycles, in particular for 0-cycles on any variety and for codimension 2 cycles on rationally connected 3-folds. For the latter, the motivation comes from the study of (stable) rationality for rationally connected 3-folds.