Algebraic Geometry
About the stability of the tangent bundle restricted to a curve
Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 421-426.

Let C be a smooth projective curve of genus g2 over an algebraically closed field k and let L be a line bundle on C generated by its global sections. The morphism ϕL:CP(H0(L))Pr is well-defined and ϕLTPr is the restriction to C of the tangent bundle of Pr. Sharpening a theorem by Paranjape, we show that if degL2gc(C) then ϕLTPr is semi-stable, specifying when it is also stable. We then prove the existence on many curves of a line bundle L of degree 2gc(C)1 such that ϕLTPr is not semi-stable. Finally, we completely characterize the (semi-)stability of ϕLTPr when C is hyperelliptic.

Soit L un fibré en droites engendré par ses sections globales sur une courbe projective lisse C de genre g2 sur un corps k algébriquement clos. Le fibré L définit ϕL:CP(H0(L))Pr et ϕLTPr est la restriction à la courbe C du fibré tangent de Pr. En précisant un théorème dû à Paranjape, on montre que si degL2gc(C) alors ϕLTPr est semi-stable, en disant quand il est aussi stable. De plus, on montre l'existence sur plusieurs courbes d'un fibré en droites L de degré 2gc(C)1 tel que ϕLTPr ne soit pas semi-stable. Enfin, on caractérise complètement la stabilité de ϕLTPr si C est hyperelliptique.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2008.02.006
Camere, Chiara 1

1 Laboratoire J.-A. Dieudonné UMR no 6621 du CNRS, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice, France
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Camere, Chiara. About the stability of the tangent bundle restricted to a curve. Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 421-426. doi : 10.1016/j.crma.2008.02.006. http://www.numdam.org/articles/10.1016/j.crma.2008.02.006/

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[2] Eisenbud, D.; Lange, H.; Martens, G.; Schreyer, F.O. The Clifford dimension of a projective curve, Compositio Math., Volume 72 (1989) no. 2, pp. 173-204

[3] Paranjape, K. http://www.imsc.res.in/~kapil/papers/chap1djvu/index.djvu (Ph.D. Thesis, available on)

[4] Schneider, O. Stabilité des fibrés ΛpEL et condition de Raynaud, Ann. Fac. Sci. Toulouse Math. (6), Volume 14 (2005) no. 3, pp. 515-525

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