On the order three Brauer classes for cubic surfaces (Q424135)

The Brauer--Manin obstruction provides a nice theory to explain the failure of the Hasse principle or weak approximation on algebraic varieties over number fields. The paper under review focuses on the computation of the Brauer--Manin obstruction for smooth cubic surfaces \(S\) over \(\mathbb{Q}\) such that \(\mathrm{Br}(S)/\mathrm{Br}(\mathbb{Q})\) has a nontrivial 3-torsion element. First, to compute a nontrivial 3-torsion Brauer class, the authors use the notion of Steiner trihedron coming from the classical geometry of lines on cubic surfaces. They present a method to associate a Brauer class with each ordered triplet of Galois invariant pairs of Steiner trihedra, and show that all order 3 classes arise in this way. Then, given an element of order 3 in the group \(\mathrm{Br}(S)/\mathrm{Br}(\mathbb{Q})\), a method to explicitly compute the induced local evaluation map is described. An advantage of the authors' approach, compared to that of \textit{J.-L. Colliot-Thélène; D. Kanevsky} and \textit{J.-J. Sansuc} [Lect. Notes Math. 1290, 1--108 (1987; Zbl 0639.14018)], is that it requires only a base field extension of degree at most 2. The authors also give explicit examples computed with their method.