Exemples de comptage de courbes sur les surfaces

TL;DR

This paper proves the geometric Manin's conjectures for certain degrees on surfaces with Cox rings satisfying a linearity property, especially generalized del Pezzo surfaces, under specific assumptions.

Contribution

It establishes the validity of geometric Manin's conjectures for a class of surfaces with a single relation Cox ring and linearity property, extending previous results.

Findings

Manin's conjectures hold for degrees in the dual of the effective cone

Results apply to generalized del Pezzo surfaces

Moduli space of morphisms has expected dimension

Abstract

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.