Curves in P3 with good restriction of the tangent bundle

Abstract

Restriction of a stable vector bundle on a variety to a subvariety does not always preserves stability. Here we consider this situation for the P3 the tangent bundle E of the projective space and its restriction to space curves X. We obtain a Shatz-stratification of the Hilbert scheme of all space curves by generalizing the stratification defined by S. Shatz of families of vector bundles by their Harder-Narasimhan polygon to even non reduced schemes. For space curves of small degree we can characterize the curves in the strata by properties of the curves and their embedding. Furthermore, we show that restriction of the tangent bundle of the projective space P3 to the general space curve is stable.