Daniele Agostini
Title: Knot theory for algebraic curves
Abstract: Inspired by knot theory, two embeddings of a curve in projective space are called algebraically isotopic if there is an A^1-family of embeddings that connects them. We show how to find isotopy invariants of embeddings in terms of determinantal representations for secant varieties of the curve and we apply these to hyperbolicity questions in real algebraic geometry. This is joint work with Mario Kummer and Jinhyung Park.
Marian Aprodu
Title: Hilbert series of Koszul modules with linear resonance
Abstract: I report on joint work with Gabi Farkas, Claudiu Raicu, and Alex Suciu. Koszul modules are finitely generated graded modules over polynomial algebras, that arise in geometric group theory in connection with numerical invariants of groups. The set-theoretic supports of Koszul modules called resonance loci, are obtained from linear sections of Grassmannians via incidence varieties and carry canonical scheme structures given by the annihilator. In our work, we find effective formulae for the Hilbert series of Koszul modules under certain natural geometric assumptions on the corresponding resonance loci. Specifically, we consider the case of linear resonance when all components are isotropic and the annihilator scheme structure is reduced.
Andrea Bianchi
Title: Hurwitz spaces and moduli of curves
Abstract: Starting from a symmetric group S_d, we introduce a topological space Hur(S_d) parameterising finite configurations of points in the complex plane with the extra information of an S_d-valued monodromy. Notably, an application of Riemann-Roch ensures that the moduli spaces of curves with directed punctures tend to arise, up to homotopy, as connected components of Hur(S_d). The stable homology of connected components of Hur(S_d) agrees with the homology of the double loop space of a certain space X_d. One can explicitly compute the rational cohomology of X_d, and in a range increasing with d the rational cohomology of its double loop space. Letting d go to infty, one recovers the Mumford conjecture on stable rational cohomology of moduli of curves, originally proved by Madsen-Weiss. I will sketch the argument, and I will report on a possible strategy to obtain information about the unstable cohomology of moduli of curves based on the higher syzygies of the cohomology of X_d.
Samir Canning
Title: The intersection theory of the moduli space of abelian fourfolds
Abstract: I will explain how to compute the Chow ring of the moduli space of principally polarized abelian fourfolds. I will also discuss the cycle theory of its toroidal compactifications. This is joint work in progress with Drakengren and Iribar López.
Dawei Chen
Title: Gorenstein contractions of multi-scale differentials
Abstract: In recent years, the degeneration of holomorphic differentials on curves with prescribed zero orders has become well understood: starting from Eisenbud–Harris limit linear series as a foundation, through Farkas–Pandharipande twisted canonical divisors, which describe the possible degeneration types, and culminating in the theory of multi-scale differentials of Bainbridge–Chen–Gendron–Grushevsky–Möller, in which the global residue condition determines the obstructions to smoothing. In this talk, based on joint work with Qile Chen, I will explain the conjecture on Gorenstein contractions of multi-scale differentials, originally proposed by Ranganathan–Wise and further developed by Battistella–Bozlee. Specifically, for a one-parameter degeneration, I will explain that multi-scale differentials can be contracted, level by level from the top down, to curves with Gorenstein singularities; moreover, the global residue condition corresponds to Rosenlicht’s descent criterion for differentials. Time permitting, I will also discuss related questions and applications concerning degenerations over higher-dimensional bases and compactifications of moduli spaces of differentials.
Karl Christ
Title: Geometry of Severi varieties on toric surfaces
Abstract: In this talk I will give an overview of a series of works together with Xiang He and Ilya Tyomkin on the geometry of Severi varieties of toric surfaces. The main focus will be on the question of their irreducibility, the so called Severi problem, and its applications to the irreducibility of other moduli spaces of curves.
David Eisenbud
Title: Hurwitz' problem on Weierstrass Semigroups
Abstract: In 1892 Hurwitz asked whether every numerical semigroup is Weierstrass; that is, is the semigroup of pole orders of meromorphic functions regular at all but one point. Buchweitz gave the first correct example showing that not every semigroup occurs.
Very recently, Frank Schreyer and I have found a new way to prove that a semigroup does not occur, using syzygies. We have proven that one of our examples is the smallest possible example in several senses, and found examples of every genus >= 13 except 18.
I will describe some of the history of this problem and explain the new ideas.
Aaron Landesman
Title: The algebraic geometry of the Putman-Wieland conjecture
Abstract: Suppose we are given a family of finite \'etale covers $X \to Y$ from a genus $g'$ curve to a genus $g \geq 2$ curve, so that a general genus $g$ curve appears in this family. The Putman-Wieland conjecture predicts that there is no common isogeny factor in the Jacobian of every such $X$. Based on joint work with Daniel Litt, we describe how to establish many new cases of the Putman-Wieland conjecture by relating the derivative of an associated period map to properties of semistable bundles on curves. We use these same ideas to resolve an open question of Prill, giving the first example of a Prill exceptional cover: a finite cover of a curve of genus at least 2 where every fiber is a divisor whose associated space of global sections is at least 2 dimensional.
Hannah K. Larson
Title: Independence of tautological classes and cohomological stability for strata of differentials
Abstract: Strata of differentials are natural subvarieties of moduli spaces of pointed curves M_{g,n}, defined by the condition that the sum of marked points with some chosen multiplicity are a (pluri)canonical divisor. The tautological ring of strata of differentials is the subring of their Chow ring generated by the restrictions of tautological classes on M_{g,n}. Earlier work of D. Chen explains some relations among the restrictions of tautological classes and shows that the tautological ring of strata of differentials is generated by divisor classes. In recent joint work with D. Chen, we prove that these are the only relations in degrees that are small compared to the genus and number of simple zeros. Moreover, for strata of holomorphic differentials, we prove that, when the simple zeros are unordered, all cohomology classes are tautological in degrees that are small compared to the genus and number of simple zeros.
Robert Lazarsfeld
Title: Syzygies in higher dimensions: what’s known and what's not
Abstract: I will present a breezy survey of where things stand concerning syzygies of algebraic varieties of dimensions two and higher.
Andreas Leopold Knutsen
Title: Distinguishing Brill-Noether loci.
Abstract: Brill-Noether theory has since the end of the 19th century studied linear systems on smooth projective curves (or, equivalently, compact Riemann surfaces). A degree d linear system of dimension r on a curve C, called a g^r_d, roughly corresponds to a non-degenerate morphism from C to P^r of degree d. In the moduli space M_g of curves of genus g one can define the Brill-Noether loci
M^r_{g,d} := {C\in Mg : C has a g^r_d},
which are closed subvarieties. The classical Brill-Noether-Petri theorem, proved first by D. Gieseker in 1982, states that a general smooth curve C of genus g admits a linear system of dimension r and degree d if and only if the Brill-Noether number
\rho(g,r,d) := g-(r+1)(g-d+r)>=0.
This can be restated as M^r_{g,d}=M_g if and only if \rho(g,r,d)>=0. When \rho(g,r,d)< 0 it is known that the codimension of any component of M^r_{g,d} is at most \rho(g,r,d). In general surprisingly little is known about the geometry of Brill-Noether loci when \rho(g,r,d)< 0, in particular about the containments between them. I will describe results from a joint work with Asher Auel and Richard Haburcak (arXiv:2406.19993), where we determine all the maximal Brill-Noether loci (wrt containment) in terms of numerical conditions on g,r,d and show that they are all distinct.
Margarida Melo
Title: Tropicalizing the Moduli Space of Trigonal Curves
Abstract: In recent years, it has become clear that many well-behaved algebro-geometric moduli spaces can be “tropicalized” via modular maps, allowing one to study properties of the original spaces through their tropical counterparts. A first step in this process is often to give a tropical interpretation of the combinatorial data used in the compactification. When such a relationship exists, the tropical modular interpretation can be used to investigate geometric properties of the original moduli space.
In this talk, I will present joint ongoing work with Angelina Zheng on this interplay in the case of trigonal curves. The geometry of trigonal curves is expected to reveal interesting topological properties of the moduli space of curves. However, tropical trigonal curves exhibit subtle combinatorial features that make the problem particularly interesting from the combinatorial point of view. In the talk, after describing our construction of a moduli space of tropical trigonal curves, I will explain how to use this space to compute the top weight rational cohomology of the moduli space of trigonal curves.
Jinhyung Park
Title: Determinantal ideals of secant varieties of algebraic curves
Abstract: Let C be a smooth projective complex curve of genus g embedded in projective space by the complete linear system of a very ample line bundle L. Eisenbud-Koh-Stillman proved that if deg L is at least 4g+2, then the defining ideal of C is generated by 2x2-minors of a matrix of linear forms, and they conjectured that if deg L is sufficiently large, then the defining ideal of the k-th secant variety of C is generated by (k+2)x(k+2)-minors of a matrix of linear forms. In this talk, we confirm this conjecture. The problem is reduced to a certain cohomology vanishing on the symmetric products of C, and this vanishing is also a key ingredient in proving the effective gonality theorem on syzygies of algebraic curves. This is joint work with Daniele Agostini.
Gianpietro Pirola
Title. On the maximal variation problem
Abstract: We study the maximum variation problem of moduli for linear systems associated with a very ample line bundle on smooth surfaces using Hodge and Picard-Lefschetz theory. We provide an affirmative answer for smooth projective regular surfaces with $p_g>0$. We extend this result to higher dimensions. This covers hyperplane sections of all smooth hypersurfaces in $\bP^{n+1}$ $n>1$ of degree $d>2$ with only $3-$ exceptional cases. Consequently, by a Beauville result, we establish a Lefschetz-type property for the Jacobian rings of these hypersurfaces. This work is a collaboration with Davide Bricalli.
Andres Rojas
Title: Coble type hypersurfaces and hyperkähler fourfolds
Abstract: A classical result, already observed by Coble, asserts that a genus 2 Jacobian (resp. the Kummer of a genus 3 Jacobian) can be embedded in \mathbb{P}^8 (resp. in \mathbb{P}^7) as the singular locus of a unique cubic (resp. quartic) hypersurface. We present a precise analogue of this result in the context of hyperkähler fourfolds, and discuss applications to the geometry of two moduli spaces of polarized hyperkähler fourfolds of K3-type. This is a joint work with Benedetta Piroddi, Ángel Ríos, and Jieao Song.
Sara Torelli
Title: Brill-Noether loci of pencils with prescribed ramification
Abstract: Classical Brill-Noether theory studies projective algebraic curves through maps to projective space, using the language of linear series. A natural refinement of this framework considers linear series with prescribed ramification. In this talk, I will address the problem of the dimension of Brill–Noether varieties of pencils (i.e., linear series of rank one) with prescribed ramifications. I will compare this refined theory to the classical Brill–Noether problem, and present recent joint work with Andreas Leopold Knutsen, which yields new results and improves upon the previous state of the art.
Filippo Viviani
Title: Moduli of sheaves on ribbons
Abastract: We discuss the geometry of the moduli stack/space of torsion-free sheaves of arbitrary rank on ribbons. This is a joint work with M. Savarese.
Qizheng Yin
Title: On the ring structure of the cohomology of compactified Jacobians
Abstract: It is a combinatorially difficult (albeit unsurprising) fact that different stability conditions can produce universal fine compactified Jacobians over the moduli of stable curves, with necessarily the same cohomology groups but different ring structures. I will describe a common degeneration of these ring structures to a bigraded ring which is independent of the stability condition. The construction involves the perverse filtration and a theory of Fourier transforms, and the resulting bigraded ring is given by the associated graded of the perverse filtration. I will also discuss two related situations -- Lagrangian fibration in compactified Jacobians and the compactified Jacobian of a single integral locally planar curve -- where taking the associated graded is expected to be unnecessary. In other words, in these two situations the cohomology ring of the compactified Jacobian (fibration) should naturally admit a bigrading splitting the perverse filtration multiplicatively. I will present some evidence supporting this expectation. Based on joint work with Younghan Bae, Davesh Maulik, and Junliang Shen.