Spin Curves Enumeration

Enumerative geometry aims at counting geometric objects that satisfy specific algebraic conditions.

The modern treatment of enumerative problems of curves has revealed deep connections between geometry, representation theory, and theoretical physics. These connections are established through the cohomology of moduli spaces of curves.

The description of the structure of these cohomology groups is crucial in this modern perspective, and I will use spin curves to address this issue.

Spin curves are refinements of curves carrying an extra sign, + or -, that enables the construction of new cohomology classes called spin classes. Spin classes carry rich algebraic structures that can be used to investigate enumerative problems. In the SpiCE project, I will study 2 open problems with this approach:Problem 1.

Dubrovin defined Frobenius manifolds to study deformations of enumerative problems at a cohomological level. How can we describe singular Frobenius manifolds?Problem 2.

How to enumerate curves in surfaces of general type?Spin curves have been extensively studied through techniques from several branches of mathematics. Yet, a general framework is missing. In SpiCE, I will combine my expertise in these fields to consider spin classes from a unified perspective. This novel approach will provide the means to prove structural results and enable significant progress on Problems 1&2.

Problems 1&2 revolve around 3 fundamental families of spin classes and their interplay, which frame the 3 work packages (WP) of SpiCE and their respective goals:WP A. Wittens class: prove a spin reconstruction theorem.WP B. Gromov-Witten theory: establish Virasoro constraints for spin curves.WP C.

Strata of differentials: express spin Wittens class via moduli spaces of differentials.The outcome is a framework that provides new methods to compute spin classes.

These effective techniques pave the way for long-term research on the deep algebraic structures associated with Problems 1&2.