Complete subvarieties of moduli spaces and the Prym map

We prove that in characteristic p>0 the locus of stable curves of p-rank at most f is pure of codimension g-f in the moduli space of stable curves. Then we consider the Prym map and analyze it using tautological classes. We study the locus of curves with an etale double cover of p-rank 0 in some detail. In particular, in genus 2 we obtain a formula for the number of such curves. We end with several examples illustrating our formula.