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

Let $C$ be a smooth projective curve of genus $g\geq 3$ and let $\eta$ be an odd theta characteristic on it such that $h^0(C,\eta) = 1$. Pick a point $p$ from the support of $\eta$ and consider the one-dimensional linear system $|\eta + p|$. 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 $|\eta + p|$ 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.