RUI: Compactifying Moduli Spaces of Orbits, Covers, and Curves

San Francisco State University, San Francisco CA

Investigators

Abstract

One of the most consequential advances in mathematics during the 20th century was motivated by a change in perspective: instead of studying a single mathematical object, we should broaden our scope and study how classes of objects fit together in families. To draw an analogy with ecology, this change in perspective is akin to the realization that, in order to understand the movement of a single fish in the sea, it helps a great deal to understand how that fish interacts with the other members of their school. In mathematics, the notion of a moduli space loosely refers to an entire family of objects; for example, in the fish analogy, the moduli space could refer to the entire school of fish. Moduli spaces can, themselves, be treated as a single entity, comprised of many, and we can learn about the objects we are interested in by studying the shape of the moduli space that parametrizes them. It can be especially enlightening to understand the shape of moduli spaces near their boundary, and the research supported by this NSF award is driven by the goal of understanding the shape of the boundary of a number of moduli spaces that parametrize different types of families of algebraic curves. This project provides research training opportunities for undergraduate and graduate students. The research aspects in this project fall into three interrelated categories, all with the common theme of investigating various compact moduli spaces of curves and what geometric and enumerative information can be gleaned from the structure of their boundary. In the first line of problems, the PI will study new classes of moduli spaces that can be realized as wonderful compactifications associated to certain complex reflection groups. These new moduli spaces provide a fertile testing ground for investigating the extent to which polyhedral methods can be generalized beyond toric varieties. In the second line of problems, the PI will introduce moduli spaces into the study of factorization problems in complex reflection groups. In particular, the primary objective is to study the polynomial structure of factorizations by constructing a suitable compactification of the associated moduli spaces of admissible covers. In the final line of problems, the PI will initiate a study of the tautological rings of the moduli spaces of pseudo-stable curves. These spaces provide alternative compactifications of the moduli spaces of curves that allow for curves with cuspidal singularities, instead of the usual nodal singularities, and progress in this research would lead to advances concerning the enumerative geometry of curves with cuspidal singularities. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.