The unirationality of $S_9^-$ and moduli spaces of pointed spin curves

TL;DR

This paper proves the unirationality of the moduli space of odd spin curves of genus 9, the highest known genus with such a property, using a birational approach involving moduli spaces of pointed spin curves.

Contribution

It establishes the unirationality of the genus 9 case and introduces a birational framework relating moduli spaces of different genera and pointed spin curves.

Findings

Unirationality of the moduli space of odd spin curves of genus 9.

A birational description of moduli spaces as projective bundles over quotients.

Results on the Kodaira dimension of moduli spaces of pointed spin curves.

Abstract

We show that the moduli space of odd spin curves of genus 9 is unirational. This is the highest genus for which such a result is known. This is achieved by realizing birationally the moduli space of odd spin curves of genus g<10 as a locally trivial projective bundle over a certain (finite quotient of the) moduli space of n-pointed odd stable spin curves of genus g'<g. We then present general results on the Kodaira dimension of both components of the moduli spaces of n-pointed spin curves of genus g.