Produit eulérien motivique et courbes rationnelles sur les variétés toriques

Abstract

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.

References in corpus (2)

• The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups
• Points de hauteur bornee sur les varietes de drapeaux en caracteristique finie

Cited by in corpus (13)

• Geometric Manin's Conjecture and rational curves
• Zeta functions, Grothendieck groups, and the Witt ring
• Motivic height zeta functions
• Beyond heights: slopes and distribution of rational points
• Classifying sections of del Pezzo fibrations, II
• Asymptotic behaviour of rational curves
• Motivic random variables and representation stability I: Configuration spaces
• Homotopy Spectra and Diophantine Equations
• Fonctions L d'Artin et nombre de Tamagawa motiviques
• Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups
• Rational approximations on toric varieties
• Motivic distribution of rational curves and twisted products of toric varieties
• Adams operations and power structures