On the rationality of the variety of smooth rational space curves with fixed degree and normal bundle

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Let S ~ n , a {\tilde S_{n,a}} be the variety of smooth, rational curves of degree n n in P 3 {{\mathbf {P}}_3} whose normal bundle has a factor of degree 2 n − 1 + a 2n - 1 + a and a factor of degree 2 n − 1 − a 2n - 1 - a . In this paper we prove that S ~ n , a {\tilde S_{n,a}} is rational if n − a n - a is even and a > 0 a > 0 .