The p-rank stratification of Artin-Schreier curves
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, p. 707-726

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.

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.

DOI : https://doi.org/10.5802/aif.2692
Classification:  11G15,  14H40,  14K15
Keywords: Artin-Schreier, hyperelliptic, curve, moduli, p-rank
@article{AIF_2012__62_2_707_0,
     author = {Pries, Rachel and Zhu, Hui June},
     title = {The $p$-rank stratification of Artin-Schreier curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     pages = {707-726},
     doi = {10.5802/aif.2692},
     mrnumber = {2985514},
     zbl = {1281.11062},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_2_707_0}
}
Pries, Rachel; Zhu, Hui June. The $p$-rank stratification of Artin-Schreier curves. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 707-726. doi : 10.5802/aif.2692. http://www.numdam.org/item/AIF_2012__62_2_707_0/

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