Fields of definition of rational curves of a given degree

Holmes, David; Rome, Nick

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 291-310.

Kontsevich et Manin ont donné une formule pour le nombre $N_e$ de courbes planes rationnelles de degré $e$ passant par $3e-1$ points en position générale dans ${\mathbb{P}}^2$. Lorsque les coordonnées de ces $3e-1$ points sont des nombres rationnels, l’ensemble de $N_e$ courbes rationnelles correspondant a une structure naturelle de module galoisien. Nous effectuons une étude élémentaire de cette structure et établissons un lien avec les transformations de revêtements de la fibre générique du produit des applications d’evaluation sur l’espace de modules de morphismes.

Nous étudions ensuite le comportement asymptotique du nombre de points rationnels sur les hypersurfaces de petit degré, ce qui nous permet de généraliser nos résultats en remplaçant le plan projectif par une telle hypersurface.

Kontsevich and Manin gave a formula for the number $N_e$ 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 $N_e$ 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.

Classification : 14H10, 14J70, 14N35, 11N36, 11P55, 14G05
Mots clés : Moduli spaces, Circle method, Rational curves, Hilbert irreducibility theorem

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