Ordinary and almost ordinary Prym varieties

Abstract

We study the $p$-rank stratification of the moduli space of Prym varieties in characteristic $p>0$. For arbitrary primes $p$ and $\ell$ with $\ell \ne p$ and integers $g\ge 3$ and $0\le f\le g$, the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified $\mathbb{Z}/\ell$-covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p\ge 5$ and $\ell =2$, the second theorem implies that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified double cover whose Prym has $p$-rank $f^{\prime }$ for each $\frac{g}{2}-1\le f^{\prime }\le g-2$; (these Pryms are not 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 $f$.

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