The geometry of the moduli space of odd spin curves

• 成果类型: Article
• 署名作者: Farkas, Gavril; Verra, Alessandro
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2014.180.3.3
• 发表日期: 2014
• 页码: 927-970
• 关键词: brill-noether-petri; kodaira dimension; bundles; families; divisors

摘要

The spin moduli space (S) over bar (g) is the parameter space of theta characteristics (spin structures) on stable curves of genus g. It has two connected components, (S) over bar (-)(g) and (S) over bar (+)(g), depending on the parity of the spin structure. We establish a complete birational classification by Kodaira dimension of the odd component (S) over bar (-)(g) of the spin moduli space. We show that (S) over bar (-)(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 of <(S)over bar>(-)(g) These we then use to explicitly describe the birational structure of S. instance, (S) over bar (-)(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 that (S) over bar (-)(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.