Components of moduli spaces of spin curves with the expected codimension

We prove a conjecture of Gavril Farkas claiming that for all integers r \geq 2 and g \geq \binom{r+2}{2} there exists a component of the locus \mathcal{S}^r_g of spin curves with a theta characteristic L such that h^0(L) \geq r+1 and h^0(L)\equiv r+1 (mod 2) which has codimension \binom{r+1}{2} inside the moduli space \mathcal{S}_g of spin curves of genus g.