Components of moduli spaces of spin curves with the expected codimension II

Abstract

We prove that for all integers $r\ge 2$ and $g\ge \lfloor \frac{r^2+10r+1}{4}\rfloor$ there exists a component of the locus ${\mathcal{S}}_g^r$ of spin curves with a theta characteristic $L$ such that $h^0(L)\ge r+1$ and $h^0(L)\equiv r+1(\text{mod}2)$ which has expected codimension $\left(\genfrac{}{}{0ex}{}{r+1}{2}\right)$ inside the moduli space ${\mathcal{S}}_g$ of spin curves of genus $g$.

Completely revised version. Lemma 2.10 has been corrected. The flow of the main argument has been remarkably simplified, Proposition 2.16 and Lemma 3.5 have been strengthened accordingly. The final part of the previous version concerning Gaussian maps on general curves has been split off and will be the object of a forthcoming paper