About the stability of the tangent bundle restricted to a curve

Camere, Chiara

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

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 ${\varphi }_L:C\to \mathbb{P}(H^0(L))\simeq {\mathbb{P}}^r$ et ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ est la restriction à la courbe C du fibré tangent de ${\mathbb{P}}^r$. En précisant un théorème dû à Paranjape, on montre que si $\mathrm{deg}L⩾2g-c(C)$ alors ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ 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 $2g-c(C)-1$ tel que ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ ne soit pas semi-stable. Enfin, on caractérise complètement la stabilité de ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ si C est hyperelliptique.

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 ${\varphi }_L:C\to \mathbb{P}(H^0(L))\simeq {\mathbb{P}}^r$ is well-defined and ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ is the restriction to C of the tangent bundle of ${\mathbb{P}}^r$. Sharpening a theorem by Paranjape, we show that if $\mathrm{deg}L⩾2g-c(C)$ then ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ is semi-stable, specifying when it is also stable. We then prove the existence on many curves of a line bundle L of degree $2g-c(C)-1$ such that ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ is not semi-stable. Finally, we completely characterize the (semi-)stability of ${\varphi }_L^{\ast }{T}_{{\mathbb{P}}^r}$ when C is hyperelliptic.

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