Fields of definition of rational curves of a given degree

Kontsevich and Manin gave a formula for the number Ne of rational plane curves of degree e through 3e−1 points in general position in the plane. When these 3e−1 points have coordinates in the rational numbers, the corresponding set of Ne rational curves has a natural Galois-module structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.