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 ⊂ 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 ⊂ 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.

On the Bombieri–Lang conjecture over finitely generated fields

BRESCIANI, GIULIO
2022

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 ⊂ 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 ⊂ 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.
2022
Settore MAT/03 - Geometria
Bombieri–Lang conjecture; varieties of general type over global fields
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/139850
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