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.
Categories: math.AG