Hyperelliptic curves with prescribed p-torsion

In this paper, we show that there exist families of curves ( defined over an algebraically closed field k of characteristic p > 2) whose Jacobians have interesting p-torsion. For example, for every 0 <= f <= g, we find the dimension of the locus of hyperelliptic curves of genus g with p-rank at most f. We also produce families of curves so that the p-torsion of the Jacobian of each fibre contains multiple copies of the group scheme alpha(p). The method is to study curves which admit an action by (Z/2)(n) so that the quotient is a projective line. As a result, some of these families intersect the hyperelliptic locus H-g.