On the Brauer group for the main open set in the affine space (Q1373661)

This paper deals with the calculation of the Brauer group of the complement of an affine hypersurface. More precisely, let \(k\) be an algebraically closed field f positive characteristic \(p\) and let \(f=f_1^{a_1} \dots f_m^{a_m} \in k[x_1, \dots,x_n]\) with \(f_i\) distinct irreducibles. Let \(Y_i\) (respectively \(Y\)) denote the closure of \(f_i^{-1}(0) \subset \mathbb A^n\) (respectively \(f^{-1}(0) \subset \mathbb A^n\)) in \(\mathbb P^n\). Under the assumption that some \(f_i\) is linear and that each \(Y_i\) is non-singular with trivial Brauer group and Picard group \(\mathbb Z\), the author announces a calculation (modulo \(p\)-groups) of the Brauer group of \(\mathbb P^n -Y = \mathbb A^n - f^{-1}(0)\). The calculation proceeds as follows: Let \(\overline{Y}=\prod Y_i\), let \(\pi: \overline{Y} \to Y\) be the map induced by the inclusions, let \(\sigma Y\) be the set of singular points of \(Y\) and let \(\sigma \overline{Y}=\pi^{-1}(\sigma Y)\). Then let \(l\) be the number of irreducible components of \(\sigma \overline{Y}\) and \(s\) the number of irreducible components of \(\sigma Y\). Then, modulo \(p\)-groups, \(\text{Br} (\mathbb P^n -Y)=(\mathbb Q/\mathbb Z)^r\) where \(r=l-m-s\).