no. 17-18
Algebraic Geometry
On semistable vector bundles over curves
[Sur les fibrés vectoriels semi-stables au-dessus des courbes]

Présenté par : Michel Raynaud

Biswas, Indranil1 ; Hein, Georg2 ; Hoffmann, Norbert3
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Universität Duisburg-Essen, Fachbereich Mathematik, 45117 Essen, Germany
3 Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany
Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 981-984.

Soit X une courbe projective lisse géométriquement irréductible définie sur un corps k, et soit E un fibré vectoriel sur X. E est semi-stable si et seulement s'il y a un fibré vectoriel F sur X tel que Hi(X,F⊗E)=0 pour i=0,1. Nous donnons une borne explicite pour le rang de F. La preuve utilise un résultat de Popa pour le cas où k est algébriquement clos.

Let X be a geometrically irreducible smooth projective curve defined over a field k, and let E be a vector bundle on X. Then E is semistable if and only if there is a vector bundle F on X such that Hi(X,F⊗E)=0 for i=0,1. We give an explicit bound for the rank of F. The proof uses a result of Popa for the case where k is algebraically closed.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.07.016
Biswas, Indranil 1 ; Hein, Georg 2 ; Hoffmann, Norbert 3

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Universität Duisburg-Essen, Fachbereich Mathematik, 45117 Essen, Germany
3 Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany 
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Biswas, Indranil; Hein, Georg; Hoffmann, Norbert. On semistable vector bundles over curves. Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 981-984. doi : 10.1016/j.crma.2008.07.016. http://www.numdam.org/articles/10.1016/j.crma.2008.07.016/

[1] Biswas, I.; Hein, G. Generalization of a criterion for semistable vector bundles, 2008 (preprint) | arXiv

[2] Faltings, G. Stable G-bundles and projective connections, J. Algebraic Geom., Volume 2 (1993), pp. 507-568

[3] Kunz, E. Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985

[4] Langton, S.G. Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math., Volume 101 (1975), pp. 88-110

[5] Popa, M. Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves, Duke Math. J., Volume 107 (2001), pp. 469-495

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