manuscripta math. 33, 111- 128 (1980) manuscripta mathematica 9 by Springer-Verla81980 NORMAL BUNDLES OF RATIONAL CURVES IN p3 Franco Ghione and Gianni Sacchiero W Let ~r = pr be the projective space over an algebrai- k ca• closed ground field k. Let X be a rational space cur m ve of degree n with only ordinary singularities. Since X is rational, the normal bundle~4r of X in p3 splits in X= = ~1~2 where ~l,and ~ 2 are line bundles, and we have deg ~i +deg~2= 4n-2. We consider the non-negative integer p defined by 2p = Ideg~l-deg~21. The aim of this paper is to determine all possible values of p and to describe the variety parametrizing all twisted rational curves in p3 with only ordinary singularities for a fixed degree n and fixed p. INTRODUCTION. In section I we develop the preparatory ma- terial and we obtain the bound on p: 0 ~ p ~ n -I In section 2 we study the cases p = n -I, p = n -2 and p = n -3. We prove that p = n -I if and only if X is a pla- ne curve, and that the case p = n -2 cannot occur. Hence if X is a twisted curve the only, possible values of p are (1) 0 ~ p < n-3 The case p = n -3 occurs if and only if the curve X is con- (*) The paper was supported by C.N.R., while both authors were members of GNSAGA O025-2611/80/OO33/Ol11/$O3.60 111