Restricted Tangent Bundle of Space Curves

Abstract

The purpose of this paper is to investigate the restriction of the tangent bundle of IP to a curve X ⊂ IP. The corresponding question for rational curves was investigated by L. Ramella [7] and F. Ghione, A. Iarrobino and G. Sacchiero [2] in the case of rational curves. Let us also mention that D. Laksov [6] proved that the restricted tangent bundle of a projectivly normal curve does not split unless the curve is rational. We will show the following theorem (See 3.1): Theorem In the variety of smooth connected space curves of genus g ≥ 1 and degree d ≥ g+ 3 there exists a nonempty dense open subset where the restricted tangent bundle is semistable and moreover simple if g ≥ 2 If the degree is high with respect to the genus (d > 3g), we get a postulation formula for the strata with a given Harder-Narasimhan polygon, following results of R. Hernandez [5]. In case of plane curves the situation is simpler due to Theorem If X is a smooth plane curve of degree d, the restricted tangent bundle is stable for d ≥ 3, of splitting type (3, 3) for a conic, and of splitting type (2, 1) for a line. Proof: (following D. Huybrechts) We denote by E the tangent bundle of IP 2 twisted by OIP 2(−1). We first suppose that d > 2. We use the facts: 1. E is stable, c1(E) = 1 and c2(E) = 1.