Rational curves on hypersurfaces of low degree, II

This is the second in a sequence of papers on the geometry of spaces of rational curves of degree e on a general hypersurface X subset of P-n of degree d. In Part I (J. reine angew. Math. 571 (2004), 73-106) it is proved that, if d < (n + 1)/2, then for each e the space of rational curves is irreducible, reduced and has the expected dimension. In this paper it is proved that, if d(2) + d + 1 less than or equal to n, then for each e the space of rational curves is a rationally connected variety; in particular it has negative Kodaira dimension.