Let C be a smooth projective curve of genus over an algebraically closed field k and let L be a line bundle on C generated by its global sections. The morphism is well-defined and is the restriction to C of the tangent bundle of . Sharpening a theorem by Paranjape, we show that if then is semi-stable, specifying when it is also stable. We then prove the existence on many curves of a line bundle L of degree such that is not semi-stable. Finally, we completely characterize the (semi-)stability of when C is hyperelliptic.
Soit L un fibré en droites engendré par ses sections globales sur une courbe projective lisse C de genre sur un corps k algébriquement clos. Le fibré L définit et est la restriction à la courbe C du fibré tangent de . En précisant un théorème dû à Paranjape, on montre que si alors 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 tel que ne soit pas semi-stable. Enfin, on caractérise complètement la stabilité de si C est hyperelliptique.
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Chiara Camere 1
@article{CRMATH_2008__346_7-8_421_0, author = {Chiara Camere}, title = {About the stability of the tangent bundle restricted to a curve}, journal = {Comptes Rendus. Math\'ematique}, pages = {421--426}, publisher = {Elsevier}, volume = {346}, number = {7-8}, year = {2008}, doi = {10.1016/j.crma.2008.02.006}, language = {en}, }
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/
[1] Geometry of Algebraic Curves, Springer-Verlag, 1985
[2] The Clifford dimension of a projective curve, Compositio Math., Volume 72 (1989) no. 2, pp. 173-204
[3] http://www.imsc.res.in/~kapil/papers/chap1djvu/index.djvu (Ph.D. Thesis, available on)
[4] Stabilité des fibrés et condition de Raynaud, Ann. Fac. Sci. Toulouse Math. (6), Volume 14 (2005) no. 3, pp. 515-525
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