Brauer–Manin obstructions on some Del Pezzo surfaces

Abstract

In an earlier paper [2], I gave explicit rules for calculating the Brauer group of a Del Pezzo surface of degree 3 or 4. Recent developments have made it desirable to extend these to the construction of the corresponding Azumaya algebras, so as to be able to calculate the associated Brauer–Manin obstruction to weak approximation or even to the Hasse principle. That is the main purpose of this note, which needs to be read in conjunction with [2]. The special case of diagonal cubic surfaces has already been treated in [1], though the process described there is not an algorithm in the strict sense; however, in numerical cases it is probably more efficient than the algorithm described in this note would be.