The p-torsion of curves with large p-rank

Abstract

Consider the locus of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g-1. We prove that the generic point of this locus has a-number 1 when f=g-2 and f=g-3. We include other results for curves with p-rank g-2 and g-3. For example, we show that the generic hyperelliptic curve with p-rank g-2 has a-number 1 and that the locus of curves with p-rank g-2 and a-number 2 is non-empty with codimension 3 in M_g. The proofs use degeneration to the boundary of M_g.