The moduli space of quasistable spin curves

TL;DR

This paper introduces a new compactification of the moduli space of theta characteristics, providing a modular interpretation and boundary stratification, with connections to tropical geometry and a refinement of spin tropical curves.

Contribution

It presents a novel compactification of the moduli space of theta characteristics with a detailed tropical geometric interpretation, distinct from existing spin curve moduli.

Findings

New compactification of the moduli space of theta characteristics

Tropical moduli space as a refinement of spin tropical curves

Explicit description of cone decompositions in the tropical setting

Abstract

We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.