THE p-TORSION OF CURVES WITH LARGE p-RANK

Consider the moduli space 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 every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with p-rank g - 2 has a - number 1 when p >= 3. We also show that the locus of curves with p-rank g - 2 and a - number 2 is non - empty with codimension 3 in M-g when p >= 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of M-g.