This afternoon session will be held during week 1 of the 2025 SRI in algebraic geometry, July 14-18, 2025 at Colorado State University in Fort Collins, Colorado. The talks in this session will focus on recent advances in the geometry of moduli spaces, especially the moduli of curves and abelian varieties.

Registration for the 2025 Summer Research Institute in Algebraic Geometry: Apply here

Organizers: Gavril Farkas, Samuel Grushevsky, and Isabel Vogt

 

Speakers

 

Schedule

Monday, July 14 (PATH 101)
  • 1:30-2:20pm: Martin Möller
  • 2:50-3:40pm: David Jensen
  • 4:10-5:00pm: Eric Larson
Tuesday, July 15 (PATH 101)
  • 1:30-2:20pm: Aaron Landesman
  • 2:50-3:40pm: Jinhyung Park
  • 4:10-5:00pm: Filippo Viviani
Thursday, July 17 (LCS ballroom)
  • 1:30-2:20pm: Hannah Larson
  • 2:50-3:40pm: Robert Lazarsfeld
  • 4:10-5:00pm: Samuel Grushevsky

Friday, July 18 (PATH 101)

  • 1:30-2:20pm: Margarida Melo
  • 2:50-3:40pm: Dawei Chen
  • 4:10-5:00pm: Angela Gibney
 

Titles and Abstracts

Dawei Chen
Singularities with Gm-action and moduli spaces of differentials
Abstract: We associate to each holomorphic differential on a smooth curve a Gorenstein singularity with Gm-action via a test configuration. This construction decomposes moduli spaces of differentials into versal deformation spaces with good Gm-action. As applications, we classify the singularities that arise in the nonvarying strata of differentials, compute the invariants of the singularities in the log minimal model program of Mg, and study the affine geometry as well as compactifications for related moduli spaces. This is based on joint work with Fei Yu.

Angela Gibney
Algebraic structures on moduli of curves from representations of VOAs
Abstract: The moduli spaces of Deligne-Mumford stable pointed curves are connected through tautological maps that reflect their recursive nature. Many useful algebraic constructions associated with these spaces are functorial with respect to such maps. One particularly rich source of examples comes from vector bundles of coinvariants and their duals—vector bundles of conformal blocks—defined via representations of vertex operator algebras V. I will describe three applications of this framework: positivity of divisors in genus zero and one, the construction of tensor products of V-modules, and criteria for semi-simplicity of V-module categories. While the actual constructions are quite involved, the talk will aim for a conceptual and non-technical perspective.

Samuel Grushevsky
Geometry of moduli of complex abelian varieties
Abstract: We survey some of the questions and progress on the questions on the geometry of the moduli spaces of complex abelian varieties, including their birational geometry, subvarieties, homology, ...

David Jensen
The Kodaira Dimensions of Moduli Spaces of Curves
Abstract: We discuss the birational geometry of various moduli spaces, including moduli of curves, abelian varieties, and Prym varieties. After surveying the current state of research in this area, we will focus on recent work showing that the moduli spaces of curves of genus 22 and 23, and the moduli space of Pryms in genus 13 are of general type. These results use tropical methods to resolve specific cases of the Strong Maximal Rank Conjecture. This is joint work with Gavril Farkas and Sam Payne.

Aaron Landesman
The asymptotic Picard rank conjecture
Abstract: The Picard rank conjecture predicts the vanishing of the rational Picard group of the Hurwitz space parameterizing simply branched covers of 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. Using the same homological stability results, we prove a version of Malle's conjecture over F_q(t), which predicts the number of G extensions of F_q(t), for G a finite group.

Eric Larson
The Maximal Rank Conjecture
Abstract: Curves in projective space can be described in either parametric or Cartesian equations. We begin by describing the Maximal Rank Conjecture, formulated originally by Severi in 1915, which prescribes a relationship between the "shape" of the parametric and Cartesian equations --- that is, which gives the Hilbert function of a general curve of genus g, embedded in P^r via a general linear series of degree d. We then explain how recent results on the interpolation problem can be used to prove this conjecture.

Hannah Larson
Moduli spaces of curves with polynomial point count
Abstract: In this talk, I will present recent advances in three closely related aspects of the geometry of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields. For curves of sufficiently small genus, the moduli space can more or less be "written down" and these invariants have been successfully computed (the nature of the description needed, and thereby the range of success, depends on the invariant). However, once the genus is large, results from birational geometry tell us such approaches are in fact impossible. Nevertheless, results in low genera supply essential base cases in inductive arguments. In particular, they have enabled us to give complete descriptions of the 11th and 13th cohomology groups of moduli spaces of stable curves, which in turn are key ingredients in new results about point counts over finite fields. This is joint work with Samir Canning, Sam Payne, and Thomas Willwacher.

Robert Lazarsfeld
Measures of irrationality and association
Abstract: I will survey a body of work addressing two related questions. First, given an irreducible complex projective variety X, how can one quantify “how irrational” X might be? More generally, given another variety Y of the same dimension, how can one measure “how far from birationally equivalent” are X and Y? The talk will emphasize examples and open problems.

Margarida Melo
Compactified spaces of roots over the space of curves.
Abstract: 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 joint work in progress with Thibault Poiret, where we study stratifications of nice compactifications of these spaces over the moduli space of stable curves using logarithmic geometry.

Martin Möller
Components of strata of abelian differentials
Abstract: Strata of abelian differentials are the moduli space of curves together with a differential form with prescribed orders of zeros and poles. We revisit the classification of components of strata of abelian differentials with an eye on arbitrary characteristic. As a main tool we construct a proper and smooth model over the integers away from a specific set of bad primes using Gorenstein contractions.

Jinhyung Park
Singularities and syzygies of secant varieties of smooth projective varieties
Abstract: I report on my recent joint work with Doyoung Choi, Justin Lacini, and John Sheridan concerning the geometric and algebraic properties of secant varieties. Given a smooth projective variety embedded in projective space, the k-th secant variety is defined as the Zariski closure of the union of the linear spans of k+1 distinct points. We assume that the embedding is given by the complete linear system of a sufficiently positive line bundle. About 10 years ago, Ullery and Chou-Song established that the first secant variety has normal Du Bois singularities, and about 5 years ago, in collaboration with Ein and Niu, we generalized this result to higher secant varieties of a curve and further showed that the minimal free resolutions of the section rings are as simple as possible in a first few steps. In joint with with Choi, Lacini, and Sheridan, we undertake a systematic study of secant varieties of a smooth projective variety of arbitrary dimension, employing the geometry of the Hilbert schemes of points. We precisely characterize when secant varieties have extra singularities not lying in lower secant varieties, and we extend the prior results on singularities and syzygies of secant varieties to the case where the Hilbert scheme of k+1 points is smooth.

Filippo Viviani
On the classification of compactified (universal) Jacobians
Abstract: 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.