Rational singularities and the Brauer group (Q1320199)

The authors present two examples of Brauer group calculations. First, they exhibit a local normal two dimensional domain \(R\), which is the local ring of an ordinary double point on a quartic in \(\mathbb{P}^ 3 (\mathbb{C})\) which has trivial divisor class group while its completion has class group of order 2 and the kernel \(B(K/R)\) of the map from the Brauer group of \(R\) to that of its quotient field \(K\) also has order 2. Next, they exhibit a rational complex surface \(X\) with a rational singular point \(p\) such that if \(R\) is the local ring of \(X\) at \(p\) and \(K\) its quotient field, then \(R\) has class group of order 2, its completion has class group cyclic of order 4, and \(B(K/R)\) is of order 2. The existence of such examples was predicted by Grothendieck in the 1960's; the sophisticated computations that the authors use to establish their examples explain the long interval between the prediction and discovery.