The p-rank stratification of Artin-Schreier curves

We study a moduli space ASg for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of ASg by p-rank into strata ASg.s of Artin-Schreier curves of genus g with prank exactly s. We enumerate the irreducible components of ASg,s and find their dimensions. As an application, when p = 2, we prove that every irreducible component of the moduli space of hyperelliptic k-curves with genus g and 2-rank s has dimension g - 1+s. We also determine all pairs (p, g) for which ASg is irreducible. Finally, we study deformations of Artin-Schreier curves with varying p-rank.