Hilbert schemes of rational curves on Fano hypersurfaces

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In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let X⊂ Pⁿ be a generic hypersurface of degree h. Let Rd(X, h) denote the open set of the Hilbert scheme parameterizing irreducible rational curves of degree d on X. We obtain that (1) If 4≤ h≤ n-1, Rd(X, h) is an integral, local complete intersection of dimension equation (n+1-h)d+n-4. equation (2) If furthermore (h²-n)d+h≤ 0 and h≥ 4, in addition to part (1), Rd(X, h) is also rationally connected.