On the normal bundles of smooth rational space curves

Abstract

in this note we consider smooth rational curves C of degree n in threedimensional projective space IP 3 (over a closed field of characteristic 0). To avoid trivial exceptions we shall always assume that n ~ 4 (this does not hold however for certain auxiliary curves we shall consider). Let N = N c be the normal bundle of C in IP 3. Since degel(IP3)=4, and d e g c l ( l P 0 = 2 , we have that d e g c l ( N ) = 4 n 2 . By a well-known theorem of Grothendieck the bundle N is a direct sum of two line bundles. Hence N ~ O c ( 2 n l a ) G O c ( 2 n 1 +a) for some non-negative a=a(C), which is uniquely determined by C. The question we would like to answer is an obvious one: which values of a occur? We shall show (Theorem 4 below) that a value a occurs if and only if 0_ =0, therefore Hi(C, N)=O. It follows [K, p. 150] that C represents a smooth point on the Chow variety Ch(3, 1, n) of effective cycles of dimension 1 and degree n in IP 3. Since the set of all smooth rational curves with a fixed degree is obviously connected, we see that the smooth C's represent a smooth, irreducible, 4n-dimensional (Zariski-)open subset S of Ch(3, 1, n). In a for thcoming paper [ E V ] we shall prove the following