Gaussian maps, Gieseker-Petri loci and large theta-characteristics

We analyze the stratification of the moduli space S_g of spin curves of genus g given by the dimension of the theta-characteristic. Using the relation between gaussian maps and the strata S_g^r, we construct “regular” components of S_g^r having expected codimension r(r+1)/2 inside S_g. We also relate moduli spaces of pointed curves with a moving spin structure to the classical Gieseker-Petri loci in M_g. We show that the locus of curves for which the Gieseker-Petri theorem fails for a pencil is always a divisor on M_g. Finally, we give a sufficient criterion for the injectivity of Gaussian maps of arbitrary line bundles on general curves of genus g.