Weak weak approximation for certain quadric surface bundles
Nick Rome
TL;DR
This work proves that for a class of biquadratic fourfolds $X\subset \mathbb{P}^3\times\mathbb{P}^2$ defined by $xy t_1^2 + xz t_2^2 + yz t_3^2 + F(x,y,z)t_4^2 = 0$ with a nondegenerate ternary quadratic form $F$ satisfying specific square-conditions, the Brauer--Manin obstruction is the only obstacle to weak approximation on smooth models, even when the base is higher dimensional. The authors employ Harari's fibration method (complemented by a dehomogenization trick to ensure a smooth section) and a detailed analysis of quadric surface bundles to derive unconditional results on weak approximation. They compute the Brauer group of a desingularisation, showing ${\rm Br}(\widetilde{X})/ {\rm Br}(k) \cong \mathbb{Z}/2\mathbb{Z}$ generated by $\beta = (-xz,-yz)_{k(\mathbb{P}^2)}$, with residues governed by the geometry of the coordinate lines. The obstruction set is then explicit: $X(\mathbb{A}_k)^{\mathrm{Br}}$ is controlled by archimedean places and primes above $2$, and a concrete Hassett--Pirutka--Tschinkel type example demonstrates failure of weak approximation at places in the exceptional set, illustrating a transcendental Brauer element in action. Overall, the paper extends unconditional Brauer--Manin obstruction results to higher-dimensional bases and provides new instances of weak weak approximation in this Diophantine context.
Abstract
We investigate weak approximation away from a finite set of places for a class of biquadratic fourfolds inside $\mathbb{P}^3 \times \mathbb{P}^2$, some of which appear in the recent work of Hassett--Pirutka--Tschinkel.