Brauer groups of certain affine cubic 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.