The Birational Type of the Moduli Space of Spin Curves
نویسنده
چکیده
The moduli space Sg of smooth spin curves parameterizes pairs [C, η], where [C] ∈ Mg is a curve of genus g and η ∈ Pic(C) is a theta-characteristic. The finite forgetful map π : Sg → Mg has degree 22g and Sg is a disjoint union of two connected components S+ g and S− g of relative degrees 2g−1(2g +1) and 2g−1(2g − 1) corresponding to even and odd theta-characteristics respectively. A compactification Sg of Sg overMg is obtained by considering the coarse moduli space of the stack of stable spin curves of genus g (cf. [C], [CCC] and [AJ]). The projection Sg → Mg extends to a finite branched covering π : Sg → Mg. In this paper we determine the Kodaira dimension of S+g :
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