Rational curves on hypersurfaces of low degree

We prove that for n > 2 and d > n+1/2, a general complex hypersurface X ⊂ ℙⁿ of degree d has the property that for each integer e the scheme R_e(X) parametrizing degree e, smooth rational curves on X is an integral, local complete intersection scheme of "expected" dimension (n + 1 - d)e + (n - 4). 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, ℳ̄_0,r(X, e). In particular we use the deformation theory of stable maps, properness of the stack ℳ̄_0,r(X, e), and the decomposition of ℳ̄ _0,r(X, e) described in [2]. (3) A version of Mori's bend-and-break lemma.