About the stability of the tangent bundle restricted to a curve

TL;DR

This paper investigates the stability properties of the tangent bundle restricted to a curve, providing conditions for semi-stability and stability, and characterizing these properties for hyperelliptic curves.

Contribution

It sharpens existing theorems on the stability of the restricted tangent bundle and characterizes stability conditions for hyperelliptic curves.

Findings

If deg L > 2g - c(C) - 1, then i*T is semi-stable.

Existence of line bundles L with degree 2g - c(C) - 1 where i*T is not semi-stable.

Complete characterization of (semi-)stability for hyperelliptic curves.

Abstract

Let C be a smooth projective curve with genus g>1 and Clifford index c(C) and let L be a line bundle on C generated by its global sections. The morphism i:C -->P(H^0(L))=P is well-defined and i*T is the restriction to C of the tangent bundle T of the projective space P. Sharpening a theorem by Paranjape, we show that if deg L>2g-c(C)-1 then i*T 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 i*T is not semi-stable. Finally, we completely characterize the (semi-)stability of i*T when C is hyperelliptic.