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Let O*Tp3 be the pull-back of the tangent bundle to P3 via a parametrization ik of a rational, reduced, irreducible curve C in P3 contained in an irreducible quadric surface. Since C is rational, the bundle i*Tp3 splits into the direct sum of three line bundles. In this paper we study the relationship between the degrees of the line bundles of the splitting of O*Tp3 and the geometry of the curve C. 0. Introduction. Let Tpr be the tangent bundle to the r-dimensional projective space pr over an algebraically closed field. Throughout this paper C will denote a rational, reduced, irreducible curve in pr of degree dc not contained in any hyperplane and 4: p' -l pr will be a parametrization of C. We will drop the subscript C anytime this will not lead to confusion. We consider the vector bundle 4'*Tpr on P' and, by abuse of language, we will refer to it as the tangent bundle restricted to the curve C. By a well-known theorem of Grothendieck V)*Tpr splits into the direct sum of r line bundles. The aim of this paper is to determine the decomposition of the tangent bundle restricted to a singular curve lying on a quadric surface in P3. The motivation for studying V)*Tpr comes from the relationship (suggested in [EV]) between it and the geometry of the embedded curve, and because of the information about the normal bundle, the Hilbert function, and the regularity of the curve one can derive from the splitting of 4'*Tpr. We distinguish between the cases where the quadric surface Q is singular from the ones where it is not. If C lies on a quadric cone then we will denote by a the intersection number of C with the lines of the ruling on Q outside the vertex; if C lies on a smooth quadric then (a, d a) will be its divisor class. In both cases we can assume that a < [d/2J. Our main results are the following THEOREM (0. 1). If C lies on a quadric cone, it passes through its vertex and one of the following holds: (i) a < Ld/3], (ii) Ld/3j < a and there exist two points on C of multiplicity a, (iii) Ld/3J < a and there exists a point on C of multiplicity a such that the total number of inflections at the point is a. Then O*Tpe v Opi (d + a) ED Opi (d + a) (D Op (2d 2a). Received by the editors November 4, 1985. 1980 Mathemotics Sukiect Cla&sifintion (1985 Revision). Primary 14H45. This paper is part of the author's Ph.D. thesis prepared under the supervision of Dr. D. Eisenbud (Brandeis University, May 1985). (g)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

The restricted tangent bundle of a rational curve on a quadric in 𝑃³