Tropical Methods for the Tautological Intersection Theory of the Moduli Spaces of Curves

Algebraic geometry is a broad and active area of research in mathematics. Moduli spaces, geometric objects whose points parameterize other geometric objects, are of fundamental importance both in algebraic geometry, and in connecting algebraic geometry to other areas of science. For example, the connection with physics arises from the fact that the evolution of strings in space-time may be interpreted as an appropriate measurement on a moduli space of stable maps to space-time.

The geometry of moduli space is extremely sophisticated, but it often comes with a rich recursive structure: in simple terms, more complicated moduli spaces contain within themselves a skeleton built of simpler moduli spaces. Over the last few decades this phenomenon has led to the development of several combinatorial approaches to the study of intersection theory of moduli spaces.

The main goal of this project is to develop a thorough understanding of the intersection theory of a particular class of moduli spaces, called admissible cover spaces. Admissible cover spaces provide a rich and interesting connection between algebraic geometry and representation theory of finite groups, and have significant applications to mathematical physics and mirror symmetry.

The goal will be achieved through a combination of several techniques and perspectives, including methods coming from tropical geometry, logarithmic geometry and mathematical physics. The PI, together with collaborators, will work in parallel both to further develop and to apply these techniques to the study of the structure of moduli spaces of admissible covers. This project provides research training opportunities for students.

Specific projects contributing to achieving the main goal are organized in three groups. The first group of projects explores the structure of classes of hyperelliptic curves with marked Weierstrass points and pairs of conjugate points. The aim is to generalize the notion of Cohomological Field Theory, and to exploit this structure to obtain graph formulas for these classes.

A better understanding of the structure of admissible cover loci is a tool to recover enumerative information hidden in Gromov-Witten invariants of curves. The second group of projects aims to give a solid combinatorial framework for the tautological intersection theory of a directed system of birational models of the moduli space of curves, obtained by blowing up all boundary strata (and proper transforms thereof).

Besides being of independent interest, we expect this calculus to be an important tool in understanding families of classes of admissible covers, whose intersection with the boundary is transversalized in these birational transforms. These techniques allow new perspectives on the computation of double Hurwitz numbers, and the generalization to similar enumerative geometric problems on moduli spaces of twisted log-canonical divisors.

Tropical geometry plays a fundamental role in organizing the birational modifications of the moduli space of curves that we expect to use in the study of cycles of admissible covers. The last group of projects builds on foundational work in tropical geometry that the PI conducted in collaboration with Gross and Markwig. Having defined a theory of tropical psi classes, the goal is now to establish a rigorous tropicalization statement relating algebraic and tropical classes, with the expectation that these will play a significant role in connecting algebraic and tropical intersection 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.