This paper considers the integral Brauer-Manin obstruction for open subsets of smooth del pezzo surfaces of degree 4, given by the removal of a hyperplane. Let \(A\) and \(B\) be \(5\times 5\) integer matrices, and let \(S_{A,B}\) be the intersection \(\mathbf{x}^tA\mathbf{x}=\mathbf{x}^tB\mathbf{x}=0\). When \(S_{A,B}\) is smooth of codimension 2 one is interested in the open set \(U=S_{A,B}\setminus H\), for hyperplanes \(H\) such that \(S_{A,B}\cap H\) is geometrically irreducible. In this case there can only be an integral Brauer-Manin obstruction when \[ \mathrm{BR}_1(U)/\mathrm{BR}_0(U)\not=0.\tag{*} \]
The idea of the paper is to fix a nonsingular matrix \(A\), and to count the number \(N(P)\) of matrices \(B\) of norm at most \(P\) for which (*) can hold. There are \(O(P^{15})\) possible matrices \(B\), and it is shown that the number for which there can be an integral Brauer-Manin obstruction is at most \(O_{\varepsilon}(P^{14+1/5+\varepsilon})\), for any fixed \(\varepsilon>0\).
To prove the theorem the authors use a result from their earlier work [J. Number Theory 203, 376–427 (2019; Zbl 1441.14073)], showing that one can only have (*) when the quintic form \(f(\lambda,\mu)=\mathrm{det}(\lambda A+\mu B)\) factors over \(\mathbb{Q}\). The counting question for this latter problem is then attacked using ideas of R. Dietmann [Mathematika 58, No. 1, 35–44 (2012; Zbl 1276.11184)], depending ultimately on the theorem of E. Bombieri and J. Pila [Duke Math. J. 59, No. 2, 337–357 (1989; Zbl 0718.11048)] which counts integer points on algebraic curves.