The authors determine the distribution of the Brauer group and the frequency of the Brauer–Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree 4 over \(\mathbb{Q}\). Del Pezzo surfaces are geometrically rational, but their arithmetic over non-algebraically closed fields is still not fully understood. Those of degree \(\geq5\) that contain a rational point are rational over the ground field, and are known to satisfy the Hasse principle.
Del Pezzo surfaces of degree 4 are therefore the first non-trivial examples of surfaces where both the Hasse principle and weak approximation can fail, and a conjecture by Colliot-Thélène and Sansuc predicts that for these surfaces, all such failures are explained by a Brauer–Manin obstruction. This conjecture is proven conditionally for general del Pezzo surfaces of degree 4 with Brauer group equal to Br \(\mathbb{Q}\) [O. Wittenberg, Intersections de deux quadriques et pinceaux de courbes de genre 1. Berlin: Springer (2007; Zbl 1122.14001)], and unconditionally for del Pezzo surfaces of BSD type [A. Várilly-Alvarado and B. Viray, Adv. Math. 255, 153–181 (2014; Zbl 1329.14051)] and for surfaces with a conic bundle structure J.-L. Colliot-Thélène [Prog. Math. 91, 43–55 (1990; Zbl 0731.14033)] and [P. Salberger, C. R. Acad. Sci., Paris, Sér. I 303, 273–276 (1986; Zbl 0612.14038)])
In this paper the authors tackle the family \(\mathcal{F}\) of del Pezzo surfaces \(X_{\textbf{a}}\) of degree 4 in \(\mathbb{P}^4_{\mathbb{Q}}\) given by \begin{align*} x_0x_1-x_2x_3&=0,\\ a_0x_0^2+a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2&=0, \end{align*} where \(\textbf{a}=(a_0,\ldots,a_4)\in\mathbb{Z}^5_{\text{prim}}\) is a vector of coprime integers with \((a_0a_1-a_2a_3)\prod_{i=0}^4a_i\neq0\). The surfaces in this family have two conic bundles, hence all failures of the Hasse principle and weak approximation come from a Brauer–Manin obstruction.
The authors show that a positive proportion of surfaces in \(\mathcal{F}\) has an adelic point (Theorem 1.1), and that for a surface \(X_{\textbf{a}}\in\mathcal{F}\), the order of Br \(X_{\textbf{a}}/\)Br \(\mathbb{Q}\), which for these surfaces is either 1, 2, or 4, is almost always 2 (Theorem 1.2). This is used to prove that the number of surfaces \(X_{\textbf{a}}\in\mathcal{F}\) with \(|\textbf{a}|\leq B\) that fail the Hasse principle is \(\ll B^{9/2}\) as \(B\) goes to infinity (Theorem.1.3), as is the number of these surfaces that satisfy weak approximation (Theorem. 1.4). This leads to the conclusion that 100 percent of the surfaces in \(\mathcal{F}\) satisfy the Hasse principle but fail weak approximation.
The results are obtained by first using classical techniques for computing the Brauer group of surfaces with a conic bundle. A careful analysis of the equations then shows for which surfaces in \(X_a\in\mathcal{F}\) the order of Br \(X_a\)/Br \(\mathbb{Q}\) equals 1, 2, or 4, and counts are given in each case. Using a thorough analysis of local densities, the proportion of surfaces that have an adelic point is computed. Theorem 1.2 is used to reduce the proofs of Theorems 1.3 and 1.4 to considering only those surfaces with Br \(X_a\)/Br \(\mathbb{Q}\) of order 2, and for these surfaces the authors find an explicit generator for the quotient. While general methods [M. Bright, Manuscr. Math. 157, No. 3–4, 529–550 (2018; Zbl 1444.11141)], [M. J. Bright et al., Compos. Math. 152, No. 7, 1435–1475 (2016; Zbl 1348.14067)] to compute the frequencies in Theorems 1.3 and 1.4 are closely followed, this explicit description of the Brauer groups allows a power saving of \(B^{1/2}\) compared to the results these general methods would give.