Persistence of the Brauer-Manin obstruction on cubic surfaces (Q6043372)

In the article under review, the authors study the Cassels and Swinnerton-Dyer conjecture for a smooth cubic surface \(X\) over a global field \(k\). Recall, that the conjecture predicts that \(X\) has a rational point if and only if it admits a degree \(1\) zero cycle. (See [\textit{D. R. Coray}, Acta Arith. 30, 267--296 (1976; Zbl 0294.14012)] for more details and discussion.) In the present article, the authors prove that if \(L/k\) is a finite extension with degree relatively prime to \(3\), then the Brauer set \(X(\mathbb{A}_L)^{\operatorname{Br}}\) is non-empty if and only if the Brauer set \(X(\mathbb{A}_k)^{\operatorname{Br}}\) is non-empty. As a consequence of this result, the authors deduce that if the Brauer-Manin obstruction is the only obstruction to the local-to-global principle for \(X\), then \(X\) has a \(k\)-rational point if and only if it admits a degree \(1\) zero cycle. A key technical step in the proof of the authors' main result is a refined Bertini type theorem that applies to embeddings of del Pezzo surfaces over characteristic \(2\) global fields.