Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - TropicalModuli (Foundations and applications of tropical moduli theory)

Teaser

One of the central ideas of modern geometry is to not only study a geometric object by itself but also to understand it as naturally sitting in a moduli space parametrizing objects of the same type together with their degenerations. In fact, just like a prism disperses light...

Summary

One of the central ideas of modern geometry is to not only study a geometric object by itself but also to understand it as naturally sitting in a moduli space parametrizing objects of the same type together with their degenerations. In fact, just like a prism disperses light into a spectrum of colors, in many cases moduli spaces allow us to access many a priori hidden properties of the original object.

Tropical geometry is the geometry of the combinatorial objects associated to degenerations and compactifications of algebraic (or analytic) varieties. This explains why, as in algebraic geometry, the tropical geometry of moduli spaces is one of the richest and most fundamental parts of this field, with many of the features of tropical geometry only being visible through the prism of moduli spaces.

The experienced researcher (in following referred to as the PI) proposed to systematically develop the foundations of tropical moduli theory, using the new stack-theoretic methods developed by him and his collaborators R. Cavalieri, M. Chan, and J. Wise, and to investigate applications to other parts of mathematics using techniques from logarithmic geometry in the sense Fontaine-Kato-Illusie.

During the fellowship the PI intended to focus on the following three types of moduli spaces (largely out of reach without the use of tropical stacks), and to explore applications to classical problems in arithmetic and algebraic geometry:

(1) the universal Picard variety, with a view towards applications to theta characteristics and spin structures, Prym varieties, and Brill-Noether theory;

(2) the moduli space of k-differentials, with a focus towards a solution of Eliashberg\'s problem (in the case k=0) and the study of compactifications of strata of abelian differentials (in the case k=1) and of quadratic differentials (in the case k=2); and

(3) the moduli space of G-admissible covers using the theory of graphs of groups.

Work performed

The PI has accepted an offer for a position as a Junior Professor at Goethe University Frankfurt am Main. That is why he has decided to terminate his Marie-Curie Fellowship after only 6 months and many of the original goals have only been partially reached.

Let us give a quick summary of the progress so far:

Regarding (1):
- Significant progress has been made on the goal of giving a stack-theoretic treatment of the universal tropical Picard variety. The PI and his collaborators, Samouil Molcho (Hebrew University), Margarida Melo (Roma Tre University), Filippo Viviani (Roma Tre University), and Jonathan Wise (University of Colorado), are currently writing up their results and a preprint will be available shortly.

- Together with Yoav Len (Georgia Institute of Technology), the PI completed a project on the tropical and non-Archimedean geometry of Prym varieties. The foundational techniques developed in this project, allowed them to proof new upper bounds on the dimensions of Prym-Brill-Noether loci for double covers of special curves, expanding on the classical work of Welters and Bertram. A preprint, with the title “Skeletons of Prym varieties and Brill-Noether theory” is already available: arXiv:1902.09410.

Regarding (2):
- Before the start of his fellowship the PI has already completed a project on the realizability problem for tropical canonical divisors, in joint work with Martin Moeller and Annette Werner (both at Goethe University of Frankfurt). Since the PI has accepted a position at Goethe University, the PI has suspended his work on that aspect of the project until he is in Frankfurt.

- Together with Madeline Brandt (University of California, Berkeley), the PI completed a project that investigates the tropical and non-Archimedean geometry of symmetric powers of algebraic curves using the moduli space of effective divisors, the PI had constructed earlier in the project with Moeller and Werner. A preprint, titled “ Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective”, is available: arXiv:1812.08740.

- Together with Dmitry Zakharov (Central Michigan University), the PI is currently completing the final write-up on the tropical geometry of the double ramification locus. A preprint will be available soon.

Regarding (3):
The original idea for Project (3), as described in the proposal, needed significant refinement. During the fellowship the PI found several collaborators to investigate these refined questions:

- Together with Yoav Len (Georgia Institute of Technology) and Dmitry Zakharov (Central Michigan University) the PI is currently working on a classification of finite abelian covers of tropical curves in terms of a stratified cohomology group. A preprint will be available soon.

- Together with Mattia Talpo (Imperial College), the PI has outlined an approach to the theory of specializing fundamental groups from algebraic to tropical curves. This appears to have unexpected applications to tropical Hurwitz theory and the construction of logarithmic compactifications of moduli spaces of curves with Teichmueller level structures. Work on a preprint has been started, but will still take some time until completion.

Final results

In (1) the project with Molcho, Melo, Viviani, and Wise will provide the mathematical community with a flexible tool to study a multitude of question related to tropical Picard varieties universally over the moduli space of curves. The project with Len lead to surprising new upper bounds on the Prym-Brill-Noether loci for double covering of special curves.

In (2) the solution of the realizability problem for canonical divisors (with Moeller and Werner) already led to new applications in our understanding of canonically embedded tropical curves of genus 3; see e.g. arXiv:1802.02440. The project with Brandt provides us with a new point of view on the Riemann-Roch formula for tropical curves, still a major unsolved problem in the field. The project with Zakharov connects tropical geometry to the study of double ramification loci, thereby connecting two different research communities.

In (3) the classification of finite abelian covers of tropical curves (with Len and Zakharov) significantly extends everything we know about (abelian) covers of tropical curves. One can think of it as a first step towards a tropical version of abelian class field theory. The results on the specialization of fundamental groups (with Talpo) appear to lead to a deeper understanding of the tropical geometry of many moduli spaces that were out of reach with the previously available techniques.