The Chow ring of ${\mathcal{S}}_5^{-}$ is tautological

Abstract

The moduli spaces ${\mathcal{S}}_g^{-}$ parametrise odd spin curves of genus $g$. These are pairs $[C,\eta ]$ where $C$ is a smooth genus $g$ curve of and $\eta$ is a line bundle on $C$ such that ${\eta }^{\otimes 2}={\omega }_C$ and $h^0(C,\eta )$ is odd. The main result of this work is the tautology of the Chow ring of ${\mathcal{S}}_5^{-}$. Our method of proof revolves around an analysis of the geometry of canonical genus 5 curves and totally tangent hyperplanes. In the course of establishing our main result, we also prove the rationality of the closely related differential stratum in ${\mathcal{M}}_{5,4}$ dominating ${\mathcal{S}}_5^{-}$.