Components of moduli spaces of spin curves with the expected codimension

Abstract

We prove a conjecture of Gavril Farkas claiming that for all integers $r \ge 2$ and $g \ge \left( {\begin{array}{c}r+2\\ 2\end{array}}\right)$ there exists a component of the locus $\mathcal {S}^r_g$ 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 codimension $\left( {\begin{array}{c}r+1\\ 2\end{array}}\right)$ inside the moduli space $\mathcal {S}_g$ of spin curves of genus $g$.