The Brauer-Manin obstruction on Del Pezzo surfaces of degree 2

Abstract

This paper explores the computation of the Brauer-Manin obstruction on Del Pezzo surfaces of degree 2, with examples coming from the class of “semi-diagonal” Del Pezzo surfaces of degree 2. It is conjectured that the failure of the Hasse principle for a broad class of varieties, including Del Pezzo surfaces, can always be explained by a nontrivial Brauer-Manin obstruction. We provide computational evidence in support of this conjecture for semi-diagonal Del Pezzo surfaces of degree 2. In addition, we determine the complete list of the possibilities for the finite abelian group H(k,PicX), where X is a Del Pezzo surface of any degree, thus completing a computation which had been previously carried out in various special cases only.