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Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups. (English) Zbl 1329.14051
The main object of the paper under review is a del Pezzo surface \(X\) of degree 4 defined over a global field \(k\) of characteristic different from 2. The authors consider the Brauer group of \(X\) and prove that if \(X\) is locally solvable (i.e., \(X(A_k)\neq\emptyset\)), then for every \(\mathcal A\in \mathrm{Br}(X)\) there exists a projection \(g: X\dasharrow \mathbb P^1\) (depending on \(\mathcal A\)), with at most two geometrically reducible fibres, such that \(\mathcal A\in g^*(\mathrm{Br} k(\mathbb P^1))\). (In common slang, this means that \(\mathrm{Br}(X)\) is vertical with respect to \(g\).) Moreover, they prove that under the same conditions, there exists \(f: X\dasharrow \mathbb P^n\) (where \(n=\dim_{\mathbb F_2}\mathrm{Br}(X)/\text{{im}} \mathrm{Br} (k)[2]\)) such that \(\mathrm{Br}(X)\) is vertical with respect to \(f\).
This allows one to deduce that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for certain quartic surfaces (under standard assumptions on the finiteness of the Tate-Shafarevich groups and Schinzel’s hypothesis).
On their way to the proof, the authors produce a practical algorithm for explicit computation of the Brauer group (which does not require computing the Galois action on exceptional curves).

14G05 Rational points
14F22 Brauer groups of schemes
Full Text: DOI
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