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We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in P n of degree 2d ≤ min(n + 4, 2n − 2) and dimension at least three is irreducible and of(More)
Given a vector bundle E on a smooth projective variety X, we can define subschemes of the Kontsevich moduli space of genus-zero stable maps M 0,0 (X, β) parameterizing maps f : P 1 → X such that the Grothendieck decomposition of f * E has a specified splitting type. In this paper, using a " compactification " of this locus, we define Gromov-Witten(More)