The geometry of the moduli space of odd spin curves

Abstract

The spin moduli spaceSg is the parameter space of theta characteristics (spin structures) on stable curves of genus g. It has two connected components, S g andS + g , depending on the parity of the spin structure. We establish a complete birational classication by Kodaira dimension of the odd componentS g of the spin moduli space. We show thatS g is uniruled for g < 12 and even unirational for g 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 ofS g . These we then use to explicitly describe the birational structure of S g . For instance, S 8 is birational to a locally trivial P 7 -bundle over the moduli space of elliptic curves with seven pairs of marked points. For g 12, we prove thatS g is a variety of general type. In genus 12, this requires the construction of a counterexample to the Slope Conjecture on eective divisors on the moduli space of stable curves of genus 12.