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

Ducrohet, Laurent

Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1641-1669.

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.

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.

Classification : 14H60
Keywords: Vector bundles, Frobenius, Prym varieties
Mot clés : fibrés vectoriels, Frobenius, variétés de Prym

[1] Beauville, A. Fibrés de rang 2 sur une courbe, fibrés déterminant et fonctions thêta, Bull. Soc. Math. France, Volume 116 (1988), pp. 431-448 | Numdam | MR | Zbl

[2] Coble, A. B. Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, Volume 10, American Mathematical Society, Providence, R.I., 1982 (Reprint of the 1929 edition) | MR

[3] Revêtements étales et groupe fondamental (SGA 1) (2003) Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)] | Zbl

[4] van Geemen, B. Schottky-Jung relations and vector bundles on hyperelliptic curves, Math. Ann., Volume 281 (1988), pp. 431-449 | DOI | MR | Zbl

[5] Gonzàlez-Dorrego, M. (16-6)-configurations and Geometry of Kummer surfaces in ${\mathbb{P}}^3$, 107, Memoirs of the American Math. Society, 1994 | MR | Zbl

[6] Griffiths, P.; Harris, J. Principles of algebraic geometry, Wiley Classics Library. John Wiley and Sons, Inc., New York, 1994 (Reprint of the 1978 original) | MR | Zbl

[7] Hartshorne, R. Algebraic geometry, Graduate Texts in Mathematics, Volume 52, Springer, New-York, 1977 | MR | Zbl

[8] Lange, H.; Pauly, C. On Frobenius-destabilized rank-2 vector bundles over curves, 2003 (arXiv : math.AG/0309456)

[9] Lange, H.; Stuhler, U. Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Zeit, Volume 156 (1977), pp. 73-83 | DOI | MR | Zbl

[10] Laszlo, Y.; Pauly, C. The action of the Frobenius map on rank 2 vector bunbles in characteristic 2, J. of Alg. Geom., Volume 11 (2002), pp. 219-243 | DOI | MR | Zbl

[11] Laszlo, Y.; Pauly, C. The Frobenius map, rank 2 vector bunbles and Kummer’s quartic surface in characteristic 2 and 3, Advances in Mathematics, Volume 185 (2004), pp. 246-269 | DOI | MR | Zbl

[12] Mochizuki, S. Foundations of $p$-adic Teichmüller theory, AMS/IP Studies in Advanced Mathematics, Providence, RI, 1999 (International Press, Cambridge, MA) | MR | Zbl

[13] Mumford, D. On equations defining abelian varieties. I, Invent. Math., Volume 1 (1966), pp. 287-354 | DOI | MR | Zbl

[14] Mumford, D. Abelian varieties, 5, Tata Institute of Fundamental Research Studies in Mathematics, Bombay, 1970 | MR | Zbl

[15] Mumford, D. Prym varieties. I, Contributions to analysis, New York Academic Press, London, 1974, pp. 325-350 | MR | Zbl

[16] Narasimhan, M. S.; Ramanan, S. Moduli of vector bundles on a compact Riemann surface, Ann. of Math., Volume 89 (1969), pp. 14-51 | DOI | MR | Zbl

[17] Narasimhan, M. S.; Ramanan, S. $2\theta$-linear systems on abelian varieties. Vector bundles on algebraic varieties, Tata Inst. Fund. Res. Stud. Math., Volume 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 415-427 (Bombay, 1984) | MR | Zbl

[18] Osserman, B. The generalized Verschiebung map for curves of genus 2, Math. Ann., Volume 336 (2006) no. 4, pp. 963-986 | DOI | MR | Zbl

[19] Osserman, B. Mochizuki’s crys-stable bundles: a lexicon and applications, Publ. Res. Inst. Math. Sci., Volume 43 (2007) no. 1, pp. 95-119 | DOI | MR | Zbl

[20] Pauly, C. Self-duality of Coble’s quartic hypersurface and applications, Michigan Math. J., Volume 50 (2002) no. 3, pp. 551-574 | DOI | MR | Zbl

[21] Raynaud, M. Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France, Volume 110 (1982), pp. 103-125 | Numdam | MR | Zbl

[22] Sekiguchi, T. On projective normality of abelian varieties. II, J. Math. Soc. Japan, Volume 29 (1977), pp. 709-727 | DOI | MR | Zbl

[23] Silverman, J. H. The arithmetic of elliptic curves, Graduate Texts in Math., Volume 106, Springer-Verlag, New-York, 1986 | MR | Zbl

[24] Zhang, B. Revêtements étales abéliens de courbes génériques et ordinarité, Ann. Fac. Sci. Toulouse, Sér. 6 (1992), pp. 133-138 | DOI | Numdam | MR | Zbl