For smooth hypersurfaces X of degree d in P^{n-1} with n > 2d(d-1)(de+1), the moduli space of degree-e maps P^2 to X is irreducible of expected dimension.
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Rational surfaces on low degree hypersurfaces
For smooth hypersurfaces X of degree d in P^{n-1} with n > 2d(d-1)(de+1), the moduli space of degree-e maps P^2 to X is irreducible of expected dimension.