On a compactification of the moduli space of the rational normal curves

For any odd $n$, we describe a smooth minimal (i.e. obtained by adding an irreducible hypersurface) compactification $\tilde S_n$ of the quasi-projective homogeneous variety $S_{n}=PGL(n+1)/SL(2)$ that parameterizes the rational normal curves in $P^n$. We show that $\tilde S_{n}$ is isomorphic to a component of the Maruyama scheme of the semi-stable sheaves on $P^n$ of rank $n$ and Chern polynomial $(1+t)^{n+2}$ and we compute its Betti numbers. In particular $\tilde S_{3}$ is isomorphic to the variety of nets of quadrics defining twisted cubics, studied by G. Ellinsgrud, R. Piene and S. Str{\o}mme (Space curves, Proc. Conf., LNM 1266).