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

Abstract

We show that, for every integer $1\le d\le 4$ and every finite set $S$ of places, there exists a degree $d$ del Pezzo surface $X$ over $\mathbb{Q}$ such that $\mathrm{Br}(X)/\mathrm{Br}(\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}$ 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 $\mathbb{Q}$.