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On the Bombieri–Lang conjecture over finitely generated fields

Giulio Bresciani

Vol. 16 (2022), No. 10, 2409–2414

The strong Bombieri–Lang conjecture postulates that, for every variety X of general type over a field k finitely generated over , there exists an open subset U X such that U(K) is finite for every finitely generated extension Kk. The weak Bombieri–Lang conjecture postulates that, for every positive dimensional variety X of general type over a field k finitely generated over , 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 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 = . Assuming the geometric Lang conjecture, we reduce the weak Bombieri–Lang conjecture to k = , too.

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Bombieri–Lang conjecture, varieties of general type over global fields
Mathematical Subject Classification
Primary: 11G35, 14G25
Received: 16 June 2021
Revised: 7 February 2022
Accepted: 4 April 2022
Published: 28 January 2023
Giulio Bresciani
Centro di Ricerca Matematica Ennio de Giorgi
Scuola Normale Superiore
Collegio Puteano