Exemples de comptage de courbes sur les surfaces

David Bourqui (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let X be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. Under a simple assumption, we show that the geometric Manin's conjectures hold for some degrees lying in the dual of the effective cone of X (in particular, for those degrees the moduli space of morphisms has the expected dimension). The result applies to a class of generalized del Pezzo surfaces which has been intensively studied in the context of the arithmetic Manin's conjecture.

Mots clés: moduli spaces of morphisms of curves; Manin's conjecture; height zeta function; global field of positive characteristic; Cox rings; singular del Pezzo surfaces

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

David Bourqui. Exemples de comptage de courbes sur les surfaces. Mathematische Annalen, 2013, 357 (4), pp.1291-1327. ⟨10.1007/s00208-013-0933-2⟩. ⟨hal-00697614⟩