On the Bertini regularity theorem for arithmetic varieties
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 601-670.

Let 𝒳 be a regular projective arithmetic variety equipped with an ample Hermitian line bundle ¯. We prove that the proportion of global sections σ with σ <1 of ¯ d whose divisor does not have a singular point on the fiber 𝒳 p over any prime pe εd tends to ζ 𝒳 (1+dim𝒳) -1 as d.

Soit 𝒳 une variété arithmétique projective régulière munie d’un fibré en droites hermitien ample ¯. On montre que la proportion des sections globales σ avec σ <1 de ¯ d dont le diviseur n’a pas de point singulier sur la fibre 𝒳 p pour tout nombre premier pe εd tend vers ζ 𝒳 (1+dim𝒳) -1 quand d.

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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
Xiaozong Wang 1

1 Morningside Center of Mathematics, Chinese Academy of Sciences Beijing, 100190, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Xiaozong Wang. On the Bertini regularity theorem for arithmetic varieties. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 601-670. doi : 10.5802/jep.191. https://jep.centre-mersenne.org/articles/10.5802/jep.191/

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