Ordinary and almost ordinary Prym varieties

TL;DR

This paper investigates the p-rank stratification of Prym varieties in positive characteristic, extending known results and constructing examples with specific p-rank properties, revealing new geometric and arithmetic insights.

Contribution

It generalizes Nakajima's result on the ordinarity of Prym varieties for unramified covers and constructs curves with Pryms of prescribed p-rank, advancing understanding of Prym moduli in characteristic p.

Findings

All Prym varieties of unramified covers of generic curves are ordinary.

Existence of curves with Prym varieties of prescribed p-rank within a certain range.

Results on the intersection properties of torsion group schemes with the theta divisor.

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 $\ell \not = p$ and integers $g \geq 3$ and $0 \leq f \leq g$, 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 $p \geq 5$ 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 $f'$ for each $\frac{g}{2}-1 \leq f' \leq g-2$; (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 $f$.