On the Brauer group of diagonal cubic surfaces (Q2922444)

The main object of the paper under review is a diagonal cubic surface \(V\) defined over a field \(k\) of characteristic zero containing all cubic roots of unity by the equation NEWLINE\[NEWLINE x^3+by^3+cz^3+dt^3=0. NEWLINE\]NEWLINE The paper contains results of two types. First, the Brauer group \(\mathrm{Br}(V)/\mathrm{Br}(k)\) is computed. This generalizes earlier computations by \textit{Yu. I. Manin} [Cubic forms. Algebra, geometry, arithmetic. Translated from Russian by M. Hazewinkel. London: North- Holland Publishing Company; New York: American Elsevier Publishing Company, Inc (1974; Zbl 0277.14014)] and \textit{J.-L. Colliot-Thélène} et al. [Lect. Notes Math. 1290, 1--108 (1987; Zbl 0639.14018)]. Second, the author is interested whether the Brauer group admits a uniform set of generators, in the sense of getting such a set by a specialization procedure from a generating set of the Brauer group \(\mathrm{Br}(V_F)/\mathrm{Br}(F)\) of the ``generic'' cubic surface \(V_F\) defined over the field \(F=k(b,c,d)\). He gives conditions for existence or non-existence of such a set.