Résumé
Soit \(X\) une surface dont l’anneau de Cox a une seule relation, laquelle vérifie en outre une certaine propriété de linéarité. Nous montrons que les conjectures de Manin géométriques valent pour certains degrés du cône effectif dual de \(X\) (notamment pour ces degrés l’espace de modules de morphismes a la dimension attendue). Le résultat s’applique à une classe de surfaces de del Pezzo généralisées qui a été intensément étudiée dans le cadre de la conjecture de Manin arithmétique.
Abstract
Some examples of curves countings on surfaces Let \(X\) be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. 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.
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Bourqui, D. Exemples de comptages de courbes sur les surfaces. Math. Ann. 357, 1291–1327 (2013). https://doi.org/10.1007/s00208-013-0933-2
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DOI: https://doi.org/10.1007/s00208-013-0933-2