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

TL;DR

This paper constructs degree d del Pezzo surfaces over rationals with specific Brauer group properties, demonstrating how the Brauer-Manin obstruction can be precisely controlled at chosen places, especially for degree four surfaces.

Contribution

It shows the existence of del Pezzo surfaces with prescribed Brauer-Manin obstructions at any finite set of places, including diagonalizability for degree four cases outside a single infinite place.

Findings

Existence of del Pezzo surfaces with prescribed Brauer-Manin obstructions.

Construction of surfaces with specific Brauer group isomorphisms.

Diagonalizability of degree four surfaces in most cases.

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 ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ 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 ${\mathbb Q}$.