Rational curves on hypersurfaces in projective n -space

We prove that for suitable dimension n and degree d, a general complex hypersurface X⊂Pn of degree d has the property that for each integer e the space ReX of degree e rational curves on X is an integral, local complete intersection scheme of dimension n+1-de+n-4 .;We also prove that for any smooth cubic hypersurface X⊂P4 , for each integer e the space ReX is an integral, local complete intersection scheme of dimension 2 e.;The techniques used in the proof include: (1) Classical results about lines on hypersurfaces including a new result about flatness of the projection map from the space of pointed lines. (2) The Kontsevich moduli space of stable maps, M0,rX,e . In particular we use the deformation theory of stable maps, the decomposition of M0,rX,e described in [Behrend-Manin96], and the fact that the coarse moduli space is a projective scheme. (3) A version of Mori's bend-and-break lemma.