Algebraic and Arithmetic Geometry Seminar
# Non-density of transcendental-rational points on Bogomolov-Tschinkel's weakly-special non-special threefolds

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Europe/Rome

Aula Seminari (Department of Mathematics)
### Aula Seminari

#### Department of Mathematics

Description

What varieties should have a dense set of rational points? Campana's conjectures from the early 2000s predict that a smooth projective variety over a number field has a potentially dense set of rational points if and only if it is special. This conjecture is in conflict with the Weakly Special Conjecture (also formulated in the early 2000s).

The latter predicts potential density holds for all weakly special varieties. However, Bogomolov-Tschinkel constructed the first example of a non-special weakly-special variety which shows the conflict.

Now, Abramovich and Varilly-Alvarado showed that Vojta's higher-dimensional abc conjecture implies that the Weakly Special Conjecture is false, so presumably Campana is right. However, up until today, we do not have a single (unconditional) example of a non-special weakly-special threefold whose rational points are not dense. In this talk, I will present joint work with Finn Bartsch and Erwan Rousseau in which we show that the "transcendental-rational points" on the threefolds of Bogomolov-Tschinkel are not dense.