On some degeneracy loci in the moduli space of pointed odd spin curves

Abstract

Let C C be a smooth projective curve of genus g ≥ 3 g\geq 3 and let η \eta be an odd theta characteristic on it such that h 0 ( C , η ) = 1 h^0(C,\eta ) = 1 . Pick a point p p from the support of η \eta and consider the one-dimensional linear system | η + p | |\eta + p| . In general this linear system is base-point free and all its ramification points are simple. The locus in the moduli space of odd spin curves is studied where the linear system | η + p | |\eta + p| fails to have this general behavior. This locus is stratified with respect to multiplicities of degeneracies; these strata are called degeneracy schemes and their geometry is explored. Conormal spaces to these schemes are described in intrinsic terms and some consequences of this are presented.