Ordinary and almost ordinary Prym varieties
Ekin Ozman, Rachel Pries

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 -rank stratification of the moduli space of Prym varieties in characteristic . For arbitrary primes and with and integers and , the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified -covers of a generic curve of genus and -rank are ordinary. Furthermore, when and , the second theorem implies that there exists a curve of genus and -rank having an unramified double cover whose Prym has -rank for each ; (these Pryms are not ordinary). Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the -torsion group scheme with the theta divisor of the Jacobian of a generic curve of genus and -rank .
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