Weak approximation on del Pezzo surfaces of degree 1

TL;DR

This paper investigates del Pezzo surfaces of degree 1, demonstrating that many fail weak approximation due to Brauer-Manin obstructions, with explicit counterexamples over number fields.

Contribution

It provides the first explicit infinite family of counterexamples to weak approximation on these surfaces using Brauer-Manin obstructions.

Findings

Infinite family of counterexamples to weak approximation

Brauer-Manin obstruction explains the failure

Explicit construction over number fields

Abstract

We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer-Manin obstruction.