The geometry of the moduli space of odd spin curves

Abstract

The spin moduli space $\overline{\mathcal{S}}_g$ is the parameter space of theta characteristics (spin structures) on stable curves of genus $g$. It has two connected components, $\overline{\mathcal{S}}_g^-$ and $\overline{\mathcal{S}}_g^+$, depending on the parity of the spin structure. We establish a complete birational classification by Kodaira dimension of the odd component $\overline{\mathcal{S}}_g^-$ of the spin moduli space. We show that $\overline{\mathcal{S}}_g^-$ is uniruled for $g<12$ and even unirational for $g\leq 8$. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus $g$, we construct new birational models of $\overline{\mathcal{S}}_g^-$. These we then use to explicitly describe the birational structure of $\overline{\mathcal{S}}_g^-$. For instance, $\overline{\mathcal{S}}_8^-$ is birational to a locally trivial $\textbf{P}^7$-bundle over the moduli space of elliptic curves with seven pairs of marked points. For $g\geq 12$, we prove that $\overline{\mathcal{S}}_g^-$ is a variety of general type. In genus $12$, this requires the construction of a counterexample to the Slope Conjecture on effective divisors on the moduli space of stable curves of genus $12$.