On the existence of ordinary and almost ordinary Prym varieties

Ekin Ozman, Rachel Pries

Published 2015-02-20Version 1

We study the relationship between the $p$-rank of a curve and the $p$-ranks of the Prym varieties of its cyclic covers in characteristic $p >0$. For arbitrary $p$, $g \geq 3$ and $0 \leq f \leq g$, we generalize a result of Nakajima by proving that the Prym varieties of all unramified cyclic degree $\ell \not = p$ covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \geq 5$, we prove that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified degree $\ell=2$ cover whose Prym is almost ordinary. Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the $\ell$-torsion group scheme with the theta divisor of the Jacobian of a generic curve $X$ of genus $g$ and $p$-rank $f$. The proofs involve geometric results about the $p$-rank stratification of the moduli space of Prym varieties.