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

[La stratification de $p$-rang des courbes d’Artin-Schreier]

Pries, Rachel ; Zhu, Hui June

Annales de l'Institut Fourier, Tome 62 (2012), p. 707-726 / Harvested from Numdam

Résumé

Nous étudions un espace de modules ${\mathcal{AS}}_g$ des courbes d’Artin Schreier de genre $g$ sur $k$, un corps algébriquement clos de caractéristique $p$. Nous étudions la stratification de ${\mathcal{AS}}_g$ par le $p$-rang, dont la strate ${\mathcal{AS}}_{g,s}$ décrit les courbes de genre $g$ et de $p$-rang $s$. On énumère les composantes irréductibles de ${\mathcal{AS}}_{g,s}$ 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 ${\mathcal{AS}}_g$ est irréductible. Finalement, nous étudions les déformations des courbes d’Artin-Schreier dont le $p$-rang varie.

Abstract

We study a moduli space ${\mathcal{AS}}_g$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$. We study the stratification of ${\mathcal{AS}}_g$ by $p$-rank into strata ${\mathcal{AS}}_{g.s}$ of Artin-Schreier curves of genus $g$ with $p$-rank exactly $s$. We enumerate the irreducible components of ${\mathcal{AS}}_{g,s}$ 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 ${\mathcal{AS}}_g$ is irreducible. Finally, we study deformations of Artin-Schreier curves with varying $p$-rank.

Détail

Publié le : 2012-01-01

MR 2985514 | Zbl 1281.11062

DOI : https://doi.org/10.5802/aif.2692
Classification:  11G15,  14H40,  14K15
Mots clés: Artin-Schreier, hyperelliptic, courbe, moduli, $p$-rank

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