Descent varieties and Brauer-Manin obstruction on diagonal cubic surfaces

Abstract: Diagonal cubic surfaces are an interesting class of varieties in the context of the Hasse principle, since they have a very simple form and the Brauer-Manin obstruction is conjectured to be the only obstruction to the Hasse principle.

In this talk, we construct torsors over diagonal cubic surfaces under a torus, which play the role of the universal torsor. We define the "type" of a torsor, and the obstruction defined by a given type. The main theorem is the equivalence between the Brauer-Manin obstruction and the obstruction defined by torsors of type $i$ that we constructed earlier. This reduces the problem of whether "the Brauer-Manin obstruction is the only one" to the validity of the Hasse principle for these torsors.

Reference:

1. Colliot-Thélène, Kanevsky, and Sansuc, Arithmétique des surfaces cubiques diagonales, Section 10(c) + Proposition 10 from (d). (English translation)

2. Colliot-Thélène and Sansuc. La descente sur les variétés rationnelles, II. (1987)

algebraic geometrynumber theory

Audience: advanced learners

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2026 topic: The descent obstruction.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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