On the Bombieri–Lang conjecture over finitely generated fields

Bresciani, Giulio

doi:10.2140/ant.2022.16.2409

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On the Bombieri–Lang conjecture over finitely generated fields

Giulio Bresciani

Vol. 16 (2022), No. 10, 2409–2414

DOI: 10.2140/ant.2022.16.2409

Abstract

The strong Bombieri–Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $ℚ$ , there exists an open subset $U \subset X$ such that $U (K)$ is finite for every finitely generated extension $K ∕ k$ . The weak Bombieri–Lang conjecture postulates that, for every positive dimensional variety $X$ of general type over a field $k$ finitely generated over $ℚ$ , the rational points $X (k)$ are not dense. Furthermore, Lang conjectured that every variety of general type $X$ over a field of characteristic $0$ contains an open subset $U \subset X$ such that every subvariety of $U$ is of general type, this statement is usually called geometric Lang conjecture.

We reduce the strong Bombieri–Lang conjecture to the case $k = ℚ$ . Assuming the geometric Lang conjecture, we reduce the weak Bombieri–Lang conjecture to $k = ℚ$ , too.

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Keywords

Bombieri–Lang conjecture, varieties of general type over global fields

Mathematical Subject Classification

Primary: 11G35, 14G25

Milestones

Received: 16 June 2021

Revised: 7 February 2022

Accepted: 4 April 2022

Published: 28 January 2023

Authors

Giulio Bresciani
Centro di Ricerca Matematica Ennio de Giorgi Scuola Normale Superiore Collegio Puteano Pisa Italy

Vol. 16, No. 10, 2022