On the Rationality of the Variety of Smooth Rational Space Curves with Fixed Degree and Normal Bundle

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