The $p$-rank stratification of Artin-Schreier curves

Pries, Rachel; Zhu, Hui June

Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 707-726.

Nous étudions un espace de modules ${\mathrm{𝒜𝒮}}_g$ des courbes d’Artin Schreier de genre $g$ sur $k$, un corps algébriquement clos de caractéristique $p$. Nous étudions la stratification de ${\mathrm{𝒜𝒮}}_g$ par le $p$-rang, dont la strate ${\mathrm{𝒜𝒮}}_{g,s}$ décrit les courbes de genre $g$ et de $p$-rang $s$. On énumère les composantes irréductibles de ${\mathrm{𝒜𝒮}}_{g,s}$ et on donne leurs dimensions. Une application, dans le cas $p=2$, est que chaque composante irréductible de l’espace de modules des courbes hyperelliptiques sur $k$ de genre $g$ et de $2$-rang $s$ est de dimension $g-1+s$. Nous déterminons toutes les paires $(p,g)$ pour lesquelles ${\mathrm{𝒜𝒮}}_g$ est irréductible. Finalement, nous étudions les déformations des courbes d’Artin-Schreier dont le $p$-rang varie.

We study a moduli space ${\mathrm{𝒜𝒮}}_g$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$. We study the stratification of ${\mathrm{𝒜𝒮}}_g$ by $p$-rank into strata ${\mathrm{𝒜𝒮}}_{g.s}$ of Artin-Schreier curves of genus $g$ with $p$-rank exactly $s$. We enumerate the irreducible components of ${\mathrm{𝒜𝒮}}_{g,s}$ and find their dimensions. As an application, when $p=2$, we prove that every irreducible component of the moduli space of hyperelliptic $k$-curves with genus $g$ and $2$-rank $s$ has dimension $g-1+s$. We also determine all pairs $(p,g)$ for which ${\mathrm{𝒜𝒮}}_g$ is irreducible. Finally, we study deformations of Artin-Schreier curves with varying $p$-rank.

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