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

Giorgio Navone, Katerina Santicola, Harry C. Shaw, Haowen Zhang

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--Thélène, 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.

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

Abstract

We construct an infinite family of quartic del Pezzo surfaces over with no quadratic points, for all primes . This answers a question of Colliot--Thélène, Creutz and Viray in the negative, which asks whether every quartic del Pezzo surface has quadratic points over fields. We exhibit a Brauer--Manin obstruction on the variety parametrising lines associated to the quartic del Pezzo surface.
Paper Structure (16 sections, 25 theorems, 100 equations)

This paper contains 16 sections, 25 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Preliminaries
  4. Notation
  5. The Brauer--Manin obstruction
  6. Quartic del Pezzo surfaces
  7. The variety
  8. Existence for
  9. The family of quartic del Pezzo surfaces
  10. Brauer--Manin obstruction on $\mathcal{G}_d$
  11. Existence of $X_d$ for $p\notin\{3,7,11\}$
  12. Existence of $X_d$ for $p\in\{7,11\}$
  13. Existence for
  14. Existence of $X_d$ for $p=3$
  15. Proof of Theorem \ref{['thm:maindp4']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $p>2$ be a prime. There exist infinitely many non-isomorphic quartic del Pezzo surfaces $X$ over $\mathbb{F}_p(t)$ such that $X(K)=\varnothing$ for all quadratic extensions $K/\mathbb{F}_p(t)$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 37 more