About the stability of the tangent bundle restricted to a curve

Abstract

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 → 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. To cite this article: C. Camere, C. R. Acad. Sci. Paris, Ser. I 346 (2008).