Hyperbola method on toric varieties

Marta Pieropan; Damaris Schindler

doi:10.5802/jep.251

Hyperbola method on toric varieties
[Méthode de l’hyperbole sur les variétés toriques]

Marta Pieropan ¹ ; Damaris Schindler ²

¹ Utrecht University, Mathematical Institute, Budapestlaan 6, 3584 CD Utrecht, the Netherlands & EPFL SB MATH CAG, Bât. MA, Station 8, 1015 Lausanne, Switzerland
² Mathematisches Institut, Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 107-157.

Résumé
Abstract

Nous développons une version très générale de la méthode de l’hyperbole qui étend la méthode connue de Blomer et Brüdern pour les produits d’espaces projectifs à des variétés toriques complètes, lisses et scindées. Nous l’utilisons pour compter les points de Campana de hauteur log-anticanonique bornée sur des $ℚ$ -variétés toriques complètes, lisses et scindées avec un bord invariant sous l’action du tore. Nous appliquons le principe de dualité forte en programmation linéaire pour montrer la compatibilité de nos résultats avec l’asymptotique conjecturée.

We develop a very general version of the hyperbola method which extends the known method by Blomer and Brüdern for products of projective spaces to complete smooth split toric varieties. We use it to count Campana points of bounded log-anticanonical height on complete smooth split toric $ℚ$ -varieties with torus invariant boundary. We apply the strong duality principle in linear programming to show the compatibility of our results with the conjectured asymptotic.

Reçu le : 2022-05-17
Accepté le : 2023-11-27
Publié le : 2023-12-21

DOI : 10.5802/jep.251

Classification : 11P21, 11A25, 11G50, 14G05, 14M25
Keywords: Hyperbola method, $m$-full numbers, Campana points, toric varieties
Mot clés : Méthode de l’hyperbole, nombres $m$-pleins, points de Campana, variétés toriques

Affiliations des auteurs :

Marta Pieropan ¹ ; Damaris Schindler ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Marta Pieropan and Damaris Schindler},
     title = {Hyperbola method on toric varieties},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {107--157},
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     year = {2024},
     doi = {10.5802/jep.251},
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Marta Pieropan; Damaris Schindler. Hyperbola method on toric varieties. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 107-157. doi : 10.5802/jep.251. https://jep.centre-mersenne.org/articles/10.5802/jep.251/

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