On the Brauer–Manin obstruction for degree-four del Pezzo surfaces
Volume 176 / 2016
Acta Arithmetica 176 (2016), 301-319
MSC: Primary 14F22; Secondary 14G25, 14J26, 11G35.
DOI: 10.4064/aa8123-8-2016
Published online: 9 November 2016
Abstract
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 $\operatorname{Br}(X)/\!\operatorname{Br}(\mathbb Q) \cong \mathbb Z/2\mathbb Z$ 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 $S = \{\infty\}$, this surface may be chosen diagonalisably over $\mathbb Q$.
