# A compactification of the moduli space of multiple-spin curves

## TL;DR

This paper introduces a new smooth compactification of the moduli space of curves with multiple spin structures, using line bundles on quasi-stable curves, and provides a detailed combinatorial and component classification.

## Contribution

It presents a novel compactification approach for multiple-spin curves, avoiding stacky curves, with a comprehensive local structure description and component classification.

## Key findings

- Constructed a smooth Deligne-Mumford compactification.
- Provided a combinatorial description of local structures.
- Classified all irreducible and connected components.

## Abstract

We construct a smooth Deligne-Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m, we give a combinatorial description of the local structure of the corresponding coarse moduli spaces. We also classify all irreducible and connected components of the resulting moduli spaces of multiple-spin curves.