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

Pries, Rachel; Zhu, Hui June

doi:10.5802/aif.2692

The

p

-rank stratification of Artin-Schreier curves
[La stratification de

p

-rang des courbes d’Artin-Schreier]

Pries, Rachel ¹ ; Zhu, Hui June ²

¹ Colorado State University Mathematics department, Weber 101 Fort Collins, CO, 80523 (USA)
² SUNY at Buffalo Mathematics department Buffalo, NY, 14260 (USA)

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

Résumé
Abstract

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

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

MR Zbl

DOI : 10.5802/aif.2692

Classification : 11G15, 14H40, 14K15
Keywords: Artin-Schreier, hyperelliptic, curve, moduli, $p$-rank
Mot clés : Artin-Schreier, hyperelliptic, courbe, moduli, $p$-rank

Affiliations des auteurs :

Pries, Rachel ¹ ; Zhu, Hui June ²

¹ Colorado State University Mathematics department, Weber 101 Fort Collins, CO, 80523 (USA)
² SUNY at Buffalo Mathematics department Buffalo, NY, 14260 (USA)

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     author = {Pries, Rachel and Zhu, Hui June},
     title = {The $p$-rank stratification of {Artin-Schreier} curves},
     journal = {Annales de l'Institut Fourier},
     pages = {707--726},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     doi = {10.5802/aif.2692},
     mrnumber = {2985514},
     zbl = {1281.11062},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2692/}
}

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Pries, Rachel; Zhu, Hui June. The $p$-rank stratification of Artin-Schreier curves. Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 707-726. doi : 10.5802/aif.2692. https://aif.centre-mersenne.org/articles/10.5802/aif.2692/

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