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On the moduli of curves with theta-characteristics. (English) Zbl 0726.14019
Let \({\mathcal M}^ r_ g\subset {\mathcal M}_ g\) be the closure of the locus of all curves of genus \(g\) having a theta-characteristic \({\mathcal L}\) (i.e. a line bundle \({\mathcal L}\) such that \({\mathcal L}^{\otimes 2}=K\), K the canonical bundle) such that \(h^ 0({\mathcal L})\geq r\) and \(h^ 0({\mathcal L})\equiv r\) mod 2.
In this paper it is shown that the tangent space to \({\mathcal M}^ r_ g\) at a point is the orthogonal to the image of the Gauss map \(\bigwedge^ 2H^ 0({\mathcal L})\to H^ 0(K)\). The author then studies \({\mathcal M}^ r_ g\) for some specific values of r and g and in particular he shows that \({\mathcal M}^ 3_ g\) has two irreducible components.

14H10 Families, moduli of curves (algebraic)
14K25 Theta functions and abelian varieties
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