On the geometry of the moduli space of spin curves

TL;DR

This paper investigates the geometric structure of the moduli space of spin curves, identifying smooth and singular loci, and proves that pluricanonical forms extend holomorphically across desingularizations.

Contribution

It precisely characterizes the smooth and canonical singularity loci of the compactified moduli space of spin curves and establishes extension properties of pluricanonical forms.

Findings

Identified the smooth locus of the moduli space of spin curves.

Determined the locus of canonical singularities.

Proved extension of pluricanonical forms to desingularizations.

Abstract

We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification \bar S_g of the moduli space S_g of spin curves, i.e., smooth curves of genus g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of \bar S_g extends holomorphically to a desingularisation of \bar S_g.