**
Beiträge zur Algebra und Geometrie **

Contributions to Algebra and Geometry

38(2), 29-436 (1997)

#
On the Stability of the Restricted Tangent Bundle of Space Curves Contained in a Quartic Surface

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E. Ballico

Dept. of Mathematics, University of Trento

38050 Povo (TN) Italy

e-mail: ballico@science.unitn.it

**Abstract:** Here we show the existence of a smooth complex projective curve $Z \subset \P^3$ of degree $d$ and genus $g$ such that $T\P^3\vert Z$ is stable for all integers $d, g$ such that (roughly) $d^2/ 9 < g < (d-1)^2/ 8$. Following a construction of Gruson and Peskine we will find such a curve $Z$ on an integral rational quartic surface with a double line. We will use in an essential way also the main properties of stable vector bundles on reduced (but reducible) curves. In this way we will obtain examples of space curves with restricted tangent bundle with high degree of stability. Then for $n = 3, 4$ and 5 we will construct curves in $\P^n$ with simple restricted tangent bundle for all invariants $(d,g)$ with (roughly) $2d\le g\le d^2/(4n-4)$ and contained in a surface of degree $2n-2$.

**Classification (MSC91):** 14H50, 14F05, 14H60, 14N05

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