About the stability of the tangent bundle restricted to a curve

Chiara Camere

doi:10.1016/j.crma.2008.02.006

Algebraic Geometry

About the stability of the tangent bundle restricted to a curve

Presented by: Michel Raynaud

Chiara Camere ¹

¹ Laboratoire J.-A. Dieudonné UMR no 6621 du CNRS, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 421-426.

Abstract (Original language)
Résumé

Let C be a smooth projective curve of genus $g ⩾ 2$ over an algebraically closed field k and let L be a line bundle on C generated by its global sections. The morphism $ϕ_{L} : C \to P (H^{0} (L)) ≃ P^{r}$ is well-defined and $ϕ_{L}^{*} T_{P^{r}}$ is the restriction to C of the tangent bundle of $P^{r}$ . Sharpening a theorem by Paranjape, we show that if $\deg L ⩾ 2 g - c (C)$ then $ϕ_{L}^{*} T_{P^{r}}$ is semi-stable, specifying when it is also stable. We then prove the existence on many curves of a line bundle L of degree $2 g - c (C) - 1$ such that $ϕ_{L}^{*} T_{P^{r}}$ is not semi-stable. Finally, we completely characterize the (semi-)stability of $ϕ_{L}^{*} T_{P^{r}}$ when C is hyperelliptic.

Soit L un fibré en droites engendré par ses sections globales sur une courbe projective lisse C de genre $g ⩾ 2$ sur un corps k algébriquement clos. Le fibré L définit $ϕ_{L} : C \to P (H^{0} (L)) ≃ P^{r}$ et $ϕ_{L}^{*} T_{P^{r}}$ est la restriction à la courbe C du fibré tangent de $P^{r}$ . En précisant un théorème dû à Paranjape, on montre que si $\deg L ⩾ 2 g - c (C)$ alors $ϕ_{L}^{*} T_{P^{r}}$ 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 $2 g - c (C) - 1$ tel que $ϕ_{L}^{*} T_{P^{r}}$ ne soit pas semi-stable. Enfin, on caractérise complètement la stabilité de $ϕ_{L}^{*} T_{P^{r}}$ si C est hyperelliptique.

Received: 2007-12-05
Accepted: 2008-02-08
Published online: 2008-03-07

DOI: 10.1016/j.crma.2008.02.006

Author's affiliations:

Chiara Camere 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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     title = {About the stability of the tangent bundle restricted to a curve},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--426},
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     number = {7-8},
     year = {2008},
     doi = {10.1016/j.crma.2008.02.006},
     language = {en},
}

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Chiara Camere. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.02.006/

References
Cited by

[1] E. Arbarello; M. Cornalba; P. Griffiths; J. Harris Geometry of Algebraic Curves, Springer-Verlag, 1985

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

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

[4] O. Schneider Stabilité des fibrés $Λ^{p} E_{L}$ et condition de Raynaud, Ann. Fac. Sci. Toulouse Math. (6), Volume 14 (2005) no. 3, pp. 515-525

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