Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials

Bertrand Deroin; Carlos Matheus

doi:10.5802/crmath.529

Géométrie algébrique

Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials
[Courbes et surfaces bi-algébriques non-linéaires dans les espaces de modules de différentielles abéliennes]

Bertrand Deroin ¹ ; Carlos Matheus ²

¹ CNRS & Université de Cergy-Pontoise (UMR CNRS 8088), 95302, Cergy-Pontoise, France
² CNRS & École Polytechnique (UMR CNRS 7640), 91128, Palaiseau, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1691-1698.

Résumé
Abstract

Les strates des espaces de modules de différentielles abéliennes sont des espaces non-homogènes possédant des structures bi-algébriques naturelles. Partiellement inspirés par le cas des espaces homogènes bi-algébriques (comme les tores, les variétés abéliennes et les variétés de Shimura), Klingler et Lerer ont récemment montré qu’une courbe bi-algébrique dans un strate d’un espace de modules de différentielles abéliennes est linéaire pourvu que la soi-disant condition $(☆)$ est satisfaite.

Dans cette note, on construit une courbe, resp. surface, bi-algébrique non-linéaire de différentielles abéliennes de genre $7$ , resp. $10$ .

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties and Shimura varieties), Klingler and Lerer recently showed that any bi-algebraic curve in a stratum of the moduli space of Abelian differentials is linear provided that the so-called condition $(☆)$ is fulfilled.

In this note, we construct a non-linear bi-algebraic curve, resp. surface, of Abelian differentials of genus $7$ , resp. $10$ .

Reçu le : 2023-03-06
Révisé le : 2023-05-31
Accepté le : 2023-06-24
Publié le : 2023-11-17

DOI : 10.5802/crmath.529

Affiliations des auteurs :

Bertrand Deroin ¹ ; Carlos Matheus ²

¹ CNRS & Université de Cergy-Pontoise (UMR CNRS 8088), 95302, Cergy-Pontoise, France
² CNRS & École Polytechnique (UMR CNRS 7640), 91128, Palaiseau, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Bertrand Deroin and Carlos Matheus},
     title = {Non-linear bi-algebraic curves and surfaces in moduli spaces of {Abelian} differentials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1691--1698},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.529},
     language = {en},
}

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Bertrand Deroin; Carlos Matheus. Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1691-1698. doi : 10.5802/crmath.529. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.529/

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