Produit eulérien motivique et courbes rationnelles sur les variétés toriques

David Bourqui (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of eulerian motivic product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.

Mots clés: Manin's conjectures; height zeta function; toric varieties; virtual motives; Euler product; pseudo-finite fields

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

David Bourqui. Produit eulérien motivique et courbes rationnelles sur les variétés toriques. Compositio Mathematica, 2009, 145 (6), pp.1360-1400. ⟨hal-00018574v2⟩