CAREER:Combinatorial Intersection Theory on Moduli Spaces of Curves

San Francisco State University, San Francisco CA

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Curves are some of the most fundamental geometric objects in mathematics; the simplest examples include such familiar shapes as the parabola and the circle, but on a deeper level, these objects play a crucial role in myriad fields of mathematics as well as the theoretical physics of string theory. Although mathematicians have studied curves for centuries, a breakthrough occurred in the late twentieth century with the advent of moduli spaces. A moduli space, roughly speaking, is the collection of all curves of a given type, and it was a groundbreaking realization that one can often more effectively understand curves by considering them in such families rather than studying them individually. In this project, the PI will undertake several sub-projects that will advance understanding of the moduli space of curves from both a theoretical and a computational standpoint, and she will initiate the study of a new variant of the moduli space that illuminates a connection between the geometry of curves, the combinatorics of polytopes, and the algebra of permutations. Alongside their intellectual merit, these projects will provide numerous avenues for student engagement: an undergraduate research program through which the PI will recruit and mentor undergraduates at her home institution of San Francisco State University (SFSU) to the Master’s level; the authoring of an algebraic geometry textbook geared toward preparing less-experienced Master’s students for research in the PI’s field; and research projects as well as community-building via which the PI will mentor Master’s-level researchers through the transition to a PhD. More technically speaking, this project is focused on two separate but interrelated lines of research. The first involves studying the intersection theory of the Deligne-Mumford moduli space of curves. Although the Chow ring of this moduli space is unwieldy in general, there is a subring known as the tautological ring that carries much of the moduli space’s geometric content while admitting an explicit set of additive generators. Continuing a longstanding research program, the PI will investigate the relations among these generators, with the long-term goal of using them to determine a formula for the Chow class of the hyperelliptic locus. The second line of research pursues a new family of moduli spaces constructed by the PI and her collaborators, which parameterize genus-zero curves with cyclic action. The motivation for these spaces arises from the fact that, in the genus-zero case, the Chow ring of the moduli space of curves admits intriguing parallels to the simpler setting of toric varieties and yet, from a birational geometry perspective, it diverges from the toric case more than was originally expected. Perhaps the most famous part of this story is Fulton’s F-conjecture, a statement about the Mori cone of the moduli space that remains unsolved. The new moduli spaces introduced by the PI and her collaborators are not toric, yet their intersection theory generalizes that of toric varieties in that it is encoded by a polytopal complex. Further investigation of these spaces will shed light on the applicability of polyhedral combinatorial methods outside the domain of toric varieties, and most ambitiously, may give a setting in which the analogue of the F-conjecture can be proven. 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.