Tropical Methods in the Study of Moduli Spaces of Families of Curves

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.