The $p$-rank stratification of the moduli space of double covers of a fixed elliptic curve

TL;DR

This paper studies the structure of the moduli space of genus g curves that double cover a fixed elliptic curve in characteristic p, showing the strata are equidimensional and constructing smooth covers for all p-ranks.

Contribution

It establishes the equidimensionality of p-rank strata and constructs smooth double covers for all p-rank values in this specific moduli space.

Findings

p-rank strata are equidimensional of expected dimension

existence of smooth double covers for all p-rank values

detailed analysis of the stratification in characteristic p

Abstract

In this paper we investigate the $p$-rank stratification of the moduli space of curves of genus $g$ that admit a double cover to a fixed elliptic curve $E$ in characteristic $p>2$. We show that the closed $p$-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of $E$ of all the possible values of the $p$-rank on this moduli space.