## Grants and Contracts Details

### Description

The study of curves is a central topic in algebraic geometry. Brill-Noether theory aims to study
the geometry of a curve by examining all of its maps to projective space, or equivalently all of its
linear series. Over the past century, the field has shifted from studying fixed to general curves –
that is, general points in the moduli space of curves Mg. This proposal focuses on outstanding
problems in the theory of curves and their moduli, and an approach to them using tropical Brill-
Noether theory.
This program has had recent success in two areas:
• A proof that the moduli spaces M22 and M23 are of general type, by establishing cases of
the Strong Maximal Rank Conjecture.
• A generalization of the Brill-Noether Theorem for general curves in the Hurwitz space Hg,k.
Building on this success, this proposal begins with the problem of computing the Kodaira dimensions
of certain moduli spaces, including Hurwitz spaces, moduli spaces of curves, and moduli
of Prym varieties. These problems serve as motivation, as each one leads to a fundamental problems
in the geometry of curves and related objects. Many of the proposed projects are essentially
combinatorial, using tropical geometry as a tool to translate geometric questions into combinatorial
ones. This includes the connection between Hurwitz spaces and Coxeter systems of type
A, as well
as the relationship between tropical linear series and the combinatorics of matroids.
2. Intellectual Merit
This proposal uses tropical methods to study problems of fundamental importance in algebraic
geometry. The principal objects of study, including Hurwitz spaces, moduli spaces of curves,
and related combinatorial structures, are of central interest not only in algebraic geometry, but
in topology, representation theory, number theory, and mathematical physics. Although these
moduli spaces have been studied extensively by generations of mathematicians, many of their basic
geometric properties remain unknown.
Recent developments in tropical geometry and combinatorics pave a path forward. These methods
have already been used to explore the Kodaira dimensions of moduli spaces and the Brill-
Noether theory of general covers, and this project will further develop these results. At the same
time, this project will push toward a deeper understanding of the combinatorics and geometry
underlying this recent progress.
3. Broader Impacts
In conjunction with this research program, the primary educational component of this proposal
concerns the Math Lab at the University of Kentucky. The lab serves as a central hub for undergraduate
research in the UK math department, where the PI has served simultaneously as a
project mentor and assistant director of communications since the lab’s inception five semesters
ago. Additional activities with significant broader impacts include mentoring postdocs, supervising
PhD students, and organizing conferences and summer schools.

Status | Active |
---|---|

Effective start/end date | 6/1/21 → 5/31/24 |

### Funding

- National Science Foundation: $292,882.00

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