Quartic del Pezzo surfaces over $\mathbb{F}_p(t)$ without quadratic points

TL;DR

This paper constructs an infinite family of quartic del Pezzo surfaces over function fields of finite fields that lack quadratic points, providing a negative answer to a question about the existence of such points.

Contribution

It demonstrates the existence of quartic del Pezzo surfaces over $\\mathbb{F}_p(t)$ without quadratic points, using Brauer--Manin obstructions, thus answering a longstanding open question.

Findings

Constructed infinite family of surfaces without quadratic points

Identified Brauer--Manin obstruction on associated line variety

Answered negatively to the question about quadratic points over $C_2$ fields

Abstract

We construct an infinite family of quartic del Pezzo surfaces over $\mathbb{F}_p(t)$ with no quadratic points, for all primes $p\neq 2$. This answers a question of Colliot--Th\'el\`ene, Creutz and Viray in the negative, which asks whether every quartic del Pezzo surface has quadratic points over $C_2$ fields. We exhibit a Brauer--Manin obstruction on the variety parametrising lines associated to the quartic del Pezzo surface.