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 $0\le f\le g$, 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 $(\text{ZZ}\mathrm{/}2{)}^n$ so that the quotient is a projective line. As a result, some of these families intersect the hyperelliptic locus ${\text{CH}}_g$.