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 and $k=\mathbb{Q}$, we give examples where the Galois invariant part descends to $\mathbb{Q}$.