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Genus one fibrations and vertical Brauer elements on del Pezzo surfaces of degree 4

Vladimir Mitankin, Cecília Salgado

TL;DR

The paper analyzes genus one fibrations arising from a special family of quartic del Pezzo surfaces with two conic bundle structures, connecting lines, Brauer groups, and Brauer–Manin obstructions. By classifying the Galois orbits of lines and applying Swinnerton-Dyer’s double-four construction, it describes when the Brauer group Br X / Br0 X is trivial, of order two, or of order four, and constructs a genus one fibration whose vertical Brauer element corresponds to a double-four. Blowing up the four base points yields a rational elliptic surface with two I4 fibres, enabling a precise study of Mordell–Weil groups and their fields of definition in relation to the Brauer group. The main results show that the Mordell–Weil group’s field of definition expands predictably with Br size: order two yields a quadratic-then-quadratic extension for a section of infinite order, while order four yields a k-rational 2-torsion section and a quadratic-extension defined full MW group. These findings illuminate the interaction between two conic bundles, vertical Brauer elements, and MW theory in this arithmetic setting, with implications for Brauer–Manin obstructions on del Pezzo surfaces.

Abstract

We consider a family of smooth del Pezzo surfaces of degree four and study the geometry and arithmetic of a genus one fibration with two reducible fibres for which a Brauer element is vertical.

Genus one fibrations and vertical Brauer elements on del Pezzo surfaces of degree 4

TL;DR

The paper analyzes genus one fibrations arising from a special family of quartic del Pezzo surfaces with two conic bundle structures, connecting lines, Brauer groups, and Brauer–Manin obstructions. By classifying the Galois orbits of lines and applying Swinnerton-Dyer’s double-four construction, it describes when the Brauer group Br X / Br0 X is trivial, of order two, or of order four, and constructs a genus one fibration whose vertical Brauer element corresponds to a double-four. Blowing up the four base points yields a rational elliptic surface with two I4 fibres, enabling a precise study of Mordell–Weil groups and their fields of definition in relation to the Brauer group. The main results show that the Mordell–Weil group’s field of definition expands predictably with Br size: order two yields a quadratic-then-quadratic extension for a section of infinite order, while order four yields a k-rational 2-torsion section and a quadratic-extension defined full MW group. These findings illuminate the interaction between two conic bundles, vertical Brauer elements, and MW theory in this arithmetic setting, with implications for Brauer–Manin obstructions on del Pezzo surfaces.

Abstract

We consider a family of smooth del Pezzo surfaces of degree four and study the geometry and arithmetic of a genus one fibration with two reducible fibres for which a Brauer element is vertical.

Paper Structure

This paper contains 13 sections, 12 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two conic bundles
  3. A special family with two conic bundles
  4. Lines and Brauer elements
  5. Conic bundles and lines
  6. Brauer elements and double fours
  7. The general case
  8. Trivial Brauer group
  9. Brauer group of order four
  10. A genus 1 fibration and vertical Brauer elements
  11. Background on elliptic surfaces
  12. Vertical elements
  13. Mordell--Weil meets Brauer

Key Result

Theorem 1.1

Let $X_\mathbf{a} \in \mathcal{F}$ and let $\mathcal{E}$ be the genus one fibration on $X_\mathbf{a}$ described in §4.2. Then the following hold.

Figures (2)

  • Figure 1: The lines on $X_\mathbf{a}$ and their intersection behaviour. The intersection points of pairs of lines are marked with $\bullet$.
  • Figure 2: The reducible fibres of the genus one fibration $\mathcal{E}$. The eight $\bullet$ denote fibre components and the four $\circ$ denote sections given by the blow up of the 4 base points.

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 21 more