Veronica Arena : TBA

Luca Battistella : Vector bundles on Olsson fans

Logarithmic geometry is a language that combines piecewise-linear and algebraic geometry, particularly useful for tracking the combinatorics of compactifications and degenerations of algebraic varieties. Olsson’s stack of logarithmic structures, and its charts provided by Artin cones, have played a fundamental role in moduli theory and enumerative geometry. Recent developments concerning stability, good moduli spaces, and logarithmic sheaf theory show the utility of considering Artin fans over a base, or Olsson fans. In joint work with Francesca Carocci and Jonathan Wise, we study their structure and vector bundles over them, generalising the theory of equivariant bundles on toric varieties. If time permits, I will discuss the relationship with Grassmannians and limit linear series that motivated our investigation.

Thomas Blomme : Correlated Gromov-Witten invariants & Multiple cover formula 

Abelian surfaces are complex tori whose enumerative invariants seem to satisfy remarkable regularity properties. The computation of their reduced Gromov-Witten invariants in the case of primitive classes has already been well studied with many complete computations by Bryan-Oberdieck-Pandharipande-Yin. A few years ago, G. Oberdieck conjectured a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. This would close the computation of GW invariants for abelian surfaces. In this second talk, we aim to explain how correlated invariants naturally show up in the decomposition formula for abelian surfaces, and how they allow to prove the multiple cover formula conjecture for many instances. This is joint work with F. Carocci.

Francesca Carocci : Correlated Gromov-Witten invariants & DR cycle formula

In this talk, we will talk about a geometric refinement for log Gromov -Witten invariants of P^1-bundles on smooth projective varieties, called correlated Gromov-Witten invariants, introduced in a joint work with T. Blomme. In order to compute them, we proved a correlated refinement of Pixton double-ramification cycle formula with target varieties. We will state the formula and try to give an idea of how it is obtained as an application of the Universal DR formula of Bae-Holmes-Pandharipande-Schmitt-Schwarz.

Aitor Iribar Lopez : TBA

Patrick Kenendy-Hunt : TBA

Navid Nabijou : Tautological projection of the Prym-Torelli locus

The moduli space of abelian varieties admits a tautological ring generated by lambda classes. In contrast to the space of stable curves, this tautological ring has a remarkably simple presentation. This allows for the construction of a canonical “tautological projection” which maps the full Chow ring onto the tautological subring, providing a left inverse to the inclusion of the latter in the former.

Given a Chow class on the moduli space of abelian varieties, it is natural to attempt to compute its tautological projection. For the Torelli locus (the locus of Jacobians) an algorithm was provided by Faber, the difficult step being integrating monomials in lambda classes on the space of stable curves.

We establish a corresponding algorithm for the Prym-Torelli locus (the locus of Prym varieties). The novel geometric content is a closed formula relating three different types of lambda classes (source, target, and Prym) on the moduli space of admissible covers, proved using Grothendieck-Riemann-Roch.

This is joint work in progress with Yoav Len and Sam Molcho.

Denis Nesterov : Hilbert-Chow crepant resolution conjecture

The conjecture in the title, proposed by Ruan, predicts that the quantum cohomology of Hilbert schemes of points on a surface is isomorphic to the orbifold cohomology of symmetric products. I will present a proof of the conjecture that relies on Fulton–MacPherson compactifications.

Sabrina Pauli : Tropical correspondence theorems for plane curve counts over arbitrary fields

We study the problem of counting rational curves of fixed degree on a toric del Pezzo surface subject to point conditions. Over algebraically closed fields, this count is invariant under the choice of point conditions. Over non-algebraically closed fields, however, the invariance fails. For real numbers, Welschinger’s groundbreaking work introduced a signed count of real curves that restores invariance.

Building on this, Levine and Kass-Levine-Solomon-Wickelgren have developed curve counts over arbitrary fields that not only generalize Welschinger’s signed counts and classical counts over algebraically closed fields, but also encode much richer arithmetic information.

In this talk I will survey these different approaches to counting rational curves with point conditions and discuss a recent joint result with A. Jaramillo Puentes, H. Markwig, and F. Röhrle. We establish a tropical correspondence theorem for curve counts over arbitrary fields, identifying the count of algebraic curves with point conditions with a weighted count of their tropical counterparts with point conditions. The latter are combinatorial objects and there are several purely combinatorial methods to find all tropical curves with point conditions.

Aaron Pixton : Cycles on universal Jacobians

Let J be the degree 0 universal Jacobian over the moduli space of smooth curves of genus g. Its Chow ring, CH(J), has an extra grading by weight (the Beauville decomposition). I will explain how to express the weight w part of CH(J) in terms of the Chow ring of a moduli space of curves with w marked points. This interpretation lets us translate ideas back and forth between cycles on universal Jacobians and cycles on moduli spaces of curves. Two consequences are a conjectural description of all tautological relations on universal Jacobians and a definition of a Fourier transform on moduli spaces of curves. I will also discuss how to extend this correspondence to compactified Jacobians over the moduli space of stable curves. This talk presents joint work with Younghan Bae.

Maximilian Schimpf : TBA

Pim Spelier : TBA

Calla Tschanz : From logarithmic Hilbert schemes to degenerations of hyperkähler varieties 

In this talk, I will discuss my previous work on constructing explicit models of logarithmic Hilbert schemes. This relates to work or Li-Wu on expanded degenerations, Gulbrandsen-Halle-Hulek on degenerations of Hilbert schemes of points and Maulik-Ranganathan on logarithmic Hilbert schemes. The constructions I consider are local. I will then explain how we globalise these in joint work with Shafi and apply them to construct minimal type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.

Angelina Zheng : The Brill-Noether rank and Martens’ theorem for tropical curves

Divisor theory on metric graphs has provided combinatorial proofs of several algebro-geometric results, most notably in Brill–Noether theory. In algebraic geometry, the dimensions of Brill–Noether loci play crucial roles, as in Martens’ theorem characterizing hyperelliptic curves. However, the tropical analogue of this result fails for metric graphs. Furthermore, the dimension of Brill–Noether loci does not vary upper semicontinuous in the moduli of tropical curves, as observed by C. M. Lim, S. Payne, and N. Potashnik. Motivated by this, D. Jensen and Y. Len conjectured that replacing the dimension with the Brill–Noether rank would yield a valid tropical analogue. However, this conjecture was later disproved by Coppens.

In this talk, based on joint work with G. Capobianco, I will discuss these counterexamples, and generalise them to a wider class of graphs. Finally, I’ll show that a suitably strengthened version of the conjecture holds for any graph.