On the Brauer-Manin obstruction for degree-four del Pezzo surfaces

We show that for every integer 1≤d≤4 and every finite set S of places, there exists a degree-d del Pezzo surface X over Q such that Br(X)/Br(Q)≅Z/2Z and the nontrivial Brauer class has a nonconstant local evaluation exactly at the places in S. For d=4, we prove that in all cases except S={∞}, this surface may be chosen diagonalisably over Q.