Ordinary and almost ordinary Prym varieties

We study the $p$-rank stratification of the moduli space of Prym varieties in characteristic $p \gt 0$. For arbitrary primes $p$ and $\ell$ with $\ell \neq p$ 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 $f^{\prime}$ for each $\frac{g}{2} - 1 \leq f^{\prime} \leq g - 2$; (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 $f$.

Keywords

Prym, curve, abelian variety, Jacobian, $p$-rank, theta divisor, torsion point, moduli space

2010 Mathematics Subject Classification

Primary 11G10, 14H10, 14H30, 14H40, 14K25. Secondary 11G20, 11M38, 14H42, 14K10, 14K15.

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The first author was partially supported by AWM-NSF Mentoring Travel Grant 2013, TUBITAK 2232 fellowship 114C126 and Bogazici University Research Grant 15B06SUP3.

The second author was partially supported by grants NSF DMS-15-02227 and NSA 131011.

Received 1 December 2016

Accepted 9 February 2018

Published 9 July 2019