  We show that, for every integer $1 \leq d \leq 4$ and every finite set $S$ of
places, there exists a degree $d$ del Pezzo surface $X$ over ${\mathbb Q}$ such
that ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ and the
Brauer-Manin obstruction works exactly at the places in $S$. For $d = 4$, we
prove that in all cases, with the exception of $S = \{\infty\}$, this surface
may be chosen diagonalizably over ${\mathbb Q}$.
