Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series

In this paper we consider some families of smooth rational curves of degree 2, 3 and 4 on a smooth Fano threefold $X$ which is a linear section of the Grassmanian $G(1,4)$ under the Plücker embedding. We prove that these families are irreducible. The proof of the irreducibility of the families of curves of degree $d$ is based on the study of degeneration of a rational curve of degree d into a curve which decomposes into an irreducible rational curve of degree $d-1$ and a projective line intersecting transversally at a point. We prove that the Hilbert scheme of curves of degree $d$ on $X$ is smooth at the point corresponding to such a reducible curve. Then calculations in the framework of deformation theory show that such a curve varies into a smooth rational curve of degree $d$. Thus, the set of reducible curves of degree $d$ of the above type lies in the closure of a unique component of the Hilbert scheme of smooth rational curves of degree $d$ on $X$. From this fact and the irreducibility of the Hilbert scheme of smooth rational curves of degree $d$ on the Grassmannian $G(1,4)$ one deduces the irreducibility of the Hilbert scheme of smooth rational curves of degree $d$ on a general Fano threefold $X$.