On the Bertini regularity theorem for arithmetic varieties

Xiaozong Wang

doi:10.5802/jep.191

On the Bertini regularity theorem for arithmetic varieties
[Autour du théorème de Bertini sur la régularité pour les variétés arithmétiques]

Xiaozong Wang ¹

¹ Morningside Center of Mathematics, Chinese Academy of Sciences Beijing, 100190, China

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 601-670.

Résumé
Abstract

Soit $𝒳$ une variété arithmétique projective régulière munie d’un fibré en droites hermitien ample $\bar{ℒ}$ . On montre que la proportion des sections globales $σ$ avec ${∥σ∥}_{\infty} < 1$ de $\bar{ℒ}^{\otimes d}$ dont le diviseur n’a pas de point singulier sur la fibre $𝒳_{p}$ pour tout nombre premier $p \leq e^{ε d}$ tend vers $ζ_{𝒳} {(1 + dim 𝒳)}^{- 1}$ quand $d \to \infty$ .

Let $𝒳$ be a regular projective arithmetic variety equipped with an ample Hermitian line bundle $\bar{ℒ}$ . We prove that the proportion of global sections $σ$ with ${∥σ∥}_{\infty} < 1$ of $\bar{ℒ}^{\otimes d}$ whose divisor does not have a singular point on the fiber $𝒳_{p}$ over any prime $p \leq e^{ε d}$ tends to $ζ_{𝒳} {(1 + dim 𝒳)}^{- 1}$ as $d \to \infty$ .

Reçu le : 2021-02-19
Accepté le : 2022-03-08
Publié le : 2022-03-30

DOI : 10.5802/jep.191

Classification : 11G35, 14G10, 14G40
Keywords: Bertini theorem, Arakelov geometry, arithmetic ampleness
Mot clés : Théorème de Bertini, géométrie d’Arakelov, amplitude arithmétique

Affiliations des auteurs :

Xiaozong Wang ¹

¹ Morningside Center of Mathematics, Chinese Academy of Sciences Beijing, 100190, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Xiaozong Wang. On the Bertini regularity theorem for arithmetic varieties. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 601-670. doi : 10.5802/jep.191. https://jep.centre-mersenne.org/articles/10.5802/jep.191/

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