The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic
Annales de l'Institut Fourier, Volume 59 (2009) no. 4, p. 1641-1669

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 3p7, 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.

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 3p7, 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.

DOI : https://doi.org/10.5802/aif.2473
Classification:  14H60
Keywords: Vector bundles, Frobenius, Prym varieties
@article{AIF_2009__59_4_1641_0,
     author = {Ducrohet, Laurent},
     title = {The Frobenius action on rank $2$ vector bundles over curves in small genus  and small characteristic},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     pages = {1641-1669},
     doi = {10.5802/aif.2473},
     mrnumber = {2566970},
     zbl = {pre05614568},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_4_1641_0}
}
Ducrohet, Laurent. The Frobenius action on rank $2$ vector bundles over curves in small genus  and small characteristic. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1641-1669. doi : 10.5802/aif.2473. http://www.numdam.org/item/AIF_2009__59_4_1641_0/

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