  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 $0 \leq f \leq g$, 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 $(\ZZ/2)^n$ so that the
quotient is a projective line. As a result, some of these families intersect
the hyperelliptic locus $\CH_g$.
