Brauer groups of certain affine cubic surfaces
Abdulmuhsin Alfaraj
TL;DR
We address the problem of determining Brauer groups for affine log K3-type surfaces $U$ obtained as the complement of a singular hyperplane section in a smooth cubic surface $X$ over a field $k$ of characteristic zero. The main method computes the Brauer group ${\rm Br}(\overline{U})$ as a $\Gamma_k$-module for all possible geometric configurations of the boundary $H$ (line+conic, irreducible singular cubic, or three lines) and then descends to ${\rm Br}(U)$ using standard exact sequences; key descriptions are given in terms of induced modules $M_d=(\mathrm{Ind}_{k(\sqrt{d})/k}\mathbb{Z})/\mathbb{Z}$. The paper provides explicit transcendental examples over $\mathbb{Q}$ in the three-line case and derives an integral Brauer–Manin obstruction for the integral Hasse principle on a concrete affine cubic surface, illustrating the arithmetic effect of the boundary geometry. These results advance understanding of Brauer groups for affine cubic surfaces and demonstrate practical obstructions to integral points via Brauer data. Overall, the work links boundary geometry, Galois modules, and Brauer groups to concrete Diophantine obstructions on affine surfaces.
Abstract
We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module for all the possible singular hyperplane sections. For the case when the hyperplane section is geometrically the union of three lines, we give explicit examples where transcendental elements of order $2$ and $3$ exist over $\mathbb{Q}$. We end with an application on the integral Brauer-Manin obstruction to the integral Hasse principle.