“Mini workshop on the moduli space of curves and related structures”
Thursday, November 10, 2022 - Room 1BC45
10:30 – 11:30: Renzo Cavalieri
11:30 – 11:45: break
11:45 – 12:45: Orsola Tommasi
14:15 – 15:15: Adam Afandi
15:15 – 15:30: break
15:30 – 16:30: Danilo Lewanski
Titles and abstracts
Renzo Cavalieri (Colorado State University)
“Pseudo-stable Hodge integrals”
The moduli space of curves (by which I mean its Deligne-Mumford compactification) is a well studied object in algebraic geometry. Mumford introduced the notion of tautological intersection theory to study a part of the intersection theory which is simple enough to be tractable, but rich enough to be meaningful. Hodge integrals are a class of tautological intersection numbers that arise from intersecting the chern classes of the Hodge bundle and of the cotangent line bundles. In the first part of the talk I will introduce all these concepts and review some “classical” structural results about Hodge integrals.
When running the MMP on the moduli space of curves, after the first wall-crossing one sees the moduli space of pseudo stable curves, which is the target of a birational regular morphism from the moduli space of curves. We investigate how the Hodge bundles on either side of this morphism are related, and how, correspondingly, there are very rich combinatorial relation between Hodge integrals and pseudo stable Hodge integrals.
This talk is based on joint work with Gallegos, Ross, Wise, Van Over and on some of Matthew Williams’ doctoral work.
Orsola Tommasi (University of Padova)
“Geometry of fine compactified Jacobians in genus 1”
The degree d universal Jacobian parametrizes degree d line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian. We focus on the case of genus 1 and obtain a combinatorial classification for fine universal compactified Jacobians, which enables us to construct new examples of them. The description we obtain for universal fine compactified Jacobians of genus 1 also yields a formula for their rational cohomology.
This is joint work with Nicola Pagani (Liverpool).
Adam Afandi (University of Münster)
“An Ehrhart Theory For Tautological Intersection Numbers”
Ehrhart polynomials are counting functions for the number of integer lattice points of dilates of bounded polyhedral objects. In this talk, I will explain how these polynomials arise when one computes tautological intersection numbers on the moduli space of pointed stable curves. A nice consequence of this connection to Ehrhart theory is that it gives a relatively concrete/simple-to-state enumerative interpretation of tautological intersection numbers. At the end of my talk, I will also talk about how one might try to generalize this Ehrhart phenomenon. The talk assumes no background in either Ehrhart theory or moduli of curves.
Danilo Lewnaski (SISSA – Trieste)
“On some hyperelliptic Hurwitz-Hodge integrals”
We address some Hodge integrals over the hyperelliptic locus. Recently Afandi computed via localisation techniques such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same result exploiting Chern classes of spin structures and relations arising from topological recursion in the sense of Eynard and Orantin. These techniques seem suitable to approach some generalisations of the problem.