The Frobenius action on rank $2$ vector bundles over curves in small genus and small characteristic

[L’action de Frobenius sur les fibrés vectoriels de rang 2 sur les courbes de petit genre en petite caractéristique]

Ducrohet, Laurent

Annales de l'Institut Fourier, Tome 59 (2009), p. 1641-1669 / Harvested from Numdam

Résumé

Soit $X$ une courbe générale, propre et lisse de genre 2 (resp. de genre 3) définie sur un corps algébriquement clos de caractéristique $p$. Lorsque $3\le p\le 7$, l’action de Frobenius sur les fibrés vectoriels semi-stable de rang 2 et de déterminant trivial est entièrement déterminée par ses restrictions aux 30 droites (resp. aux 126 surfaces de Kummer) invariantes sous l’action d’un fibré en droites d’ordre 2 sur $X$. Ces lignes (resp. ces surfaces de Kummer) sont étroitement liées aux courbes elliptiques (resp. aux surfaces abéliennes) qui apparaissent comme variétés de Prym associées aux revêtements étales doubles de $X$. Nous sommes par conséquent en mesure de calculer les équations explicites définissant l’action de Frobenius dans ces cas. Nous faisons quelques-uns de ces calculs et nous en tirons quelques conséquences géométriques.

Abstract

Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p \le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian surfaces) that appear as the Prym varieties associated to double étale coverings of $X$. We are therefore able to compute the explicit equations defining Frobenius action in these cases. We perform some of these computations and draw some geometric consequences.

Détail

Publié le : 2009-01-01

MR 2566970 | Zbl pre05614568

DOI : https://doi.org/10.5802/aif.2473
Classification:  14H60
Mots clés: fibrés vectoriels, Frobenius, variétés de Prym

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