Counting Curves Using the Topology of Moduli Spaces

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

This PI will conduct research in algebraic geometry which is the study of spaces that arise as solution sets to polynomial equations. These spaces are algebraic varieties, and they are studied both by examining the algebraic properties of the equations and the geometry of the solution sets. An important feature of algebraic geometry is that a collection of algebraic varieties (e.g. the collection of all plane conic curves) often itself is an algebraic variety, and algebraic varieties appearing in this way are called moduli spaces. The PI will study some specific moduli spaces, such as the compactified universal Jacobian and the Kontsevich moduli space of stable maps, with the goal of both better understanding them and applying their study to problems like curve counting. The grant will also support research students and the PI's outreach activities including the South Carolina Math Circle. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). After roughly 60 years of work by many mathematicians, we now have a detailed understanding of how to construct compactified Jacobians, and the PI will apply this understanding to advance algebraic geometry. The PI will study the arithmetic, geometry, and topology of moduli spaces of sheaves and then to apply those results to solve counting problems (i.e. to advance enumerative geometry). The moduli spaces the PI will focus on are moduli spaces of sheaves on singular curves or compactified Jacobians, and the project consists of two broad parts. For the first part, the PI will develop the enumerative geometry of the universal compactified Jacobian, a moduli space of sheaves on stable curves, in a manner analogous to the development of the Schubert calculus of the Grassmannian variety. For the second part, the PI will count curves arithmetically using A1-homotopy theory. 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.