Rational points on del Pezzo surfaces of degree four

TL;DR

This paper analyzes a natural family of quartic del Pezzo surfaces of degree four defined by two quadrics in $\mathbb{P}^4$, focusing on the Brauer group, Brauer--Manin obstructions, and the frequency of local solubility and weak approximation. By exploiting two conic-bundle structures, the authors give explicit descriptions of $\mathrm{Br}X_{\mathbf{a}}/\mathrm{Br}\mathbb{Q}$, including generators and residue calculations, and they compute local densities $\sigma_p$ and $\sigma_\infty$ to quantify how often the surfaces are everywhere locally soluble. They establish asymptotics for counts of surfaces with trivial, order-2, and order-4 Brauer groups, and show that order-4 cases occur infinitely often while order-2 cases dominate otherwise; they also show that all obstructions to the Hasse principle and weak approximation in this family are Brauer-Manin obstructions, with local solubility yielding a positive density. The results culminate in precise bounds: $N_1(B) \ll B^3(\log B)^4$, $N_4(B)=(60/\pi^2)B^3 + O(B^{5/2}(\log B)^2)$, and $N_2(B) \sim (32/\zeta(5))\sigma_\infty\prod_p\sigma_p B^5$, leading to a detailed picture where the Hasse principle holds universally in the locally soluble regime, but weak approximation fails in a substantial, quantifiable way. Overall, the work provides unconditional density and asymptotic results for a geometrically natural family, illustrating how explicit Brauer-group data governs diophantine phenomena on del Pezzo surfaces.

Abstract

We study 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 four over the rationals.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Proposition 3.1