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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.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14K25 Theta functions and abelian varieties
##### Keywords:
theta-characteristic; Gauss map
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##### References:
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