The geometry of the moduli space of odd spin curves

• Autores: Gavril Farkas, Alessandro Verra
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 180, Nº 3, 2014, págs. 927-970
• Idioma: inglés
• DOI: 10.4007/annals.2014.180.3.3
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• Resumen

  • The spin moduli space S ¯ ¯ g is the parameter space of theta characteristics (spin structures) on stable curves of genus g . It has two connected components, S ¯ ¯ - g and S ¯ ¯ + g , depending on the parity of the spin structure. We establish a complete birational classification by Kodaira dimension of the odd component S ¯ ¯ - g of the spin moduli space. We show that S ¯ ¯ - 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 ¯ ¯ - 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 that 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