an:04198178
Zbl 0726.14019
Nagaraj, D. S.
On the moduli of curves with theta-characteristics
EN
Compos. Math. 75, No. 3, 287-297 (1990).
00175045
1990
j
14H10 14K25
theta-characteristic; Gauss map
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.
A.Del Centina (Ferrara)