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Generators of the plane Cremona group over the field with two elements

Julia Schneider

Abstract

The plane Cremona group over the finite field $\mathbb{F}_2$ is generated by three infinite families and finitely many birational maps with small base orbits. One family preserves the pencil of lines through a point, the other two preserve the pencil of conics through four points that form either one Galois orbit of size 4, or two Galois orbits of size 2. For each family, we give a generating set that is parametrized by the rational functions over $\mathbb{F}_2$. Moreover, we describe the finitely many remaining maps and give an upper bound on the number needed to generate the Cremona group. Finally, we prove that the plane Cremona group over $\mathbb{F}_2$ is generated by involutions.

Generators of the plane Cremona group over the field with two elements

Abstract

The plane Cremona group over the finite field is generated by three infinite families and finitely many birational maps with small base orbits. One family preserves the pencil of lines through a point, the other two preserve the pencil of conics through four points that form either one Galois orbit of size 4, or two Galois orbits of size 2. For each family, we give a generating set that is parametrized by the rational functions over . Moreover, we describe the finitely many remaining maps and give an upper bound on the number needed to generate the Cremona group. Finally, we prove that the plane Cremona group over is generated by involutions.

Paper Structure

This paper contains 24 sections, 58 theorems, 90 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mori fiber spaces, Sarkisov links, and rank $r$ fibrations
  4. Finite fields
  5. General position on $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$ and some birational maps
  6. Generators of $\mathop{\mathrm{PGL}}\nolimits_2(\mathbf{k})$ and $\mathop{\mathrm{PGL}}\nolimits_3(\mathbf{k})$
  7. Groups preserving a fibration
  8. Groups preserving a pencil of conics
  9. Fixing vs. preserving a conic fibration
  10. Double sections in characteristic $2$
  11. Explicit equations for odd extensions of $\mathbb{F}_2$, and proof of Proposition \ref{['prop:ParametrizationFiberingType']}
  12. Birational Galois descent and minimal del Pezzo surfaces
  13. Del Pezzo surface of degree $8$
  14. Del Pezzo surface of degree $6$
  15. Del Pezzo surface of degree $5$
  16. ...and 9 more sections

Key Result

Proposition 1.1

Let $\mathbf{k}=\mathbb{F}_{q}$ be a finite field, and $d\in\{9,8,6,5\}$. Then $\mathcal{D}_d$ contains exactly one element $X_d$. Moreover, for each $d$ consider the following field extension $L/\mathbf{k}$, surface $Y$, and birational map $\varphi\in\mathop{\mathrm{Bir}}\nolimits(\mathbb{P}^2_L)$: Then there exists a birational morphism $\rho\colon (X_d)_L \, \begin{tikzpicture}[baseline=-.6ex]\

Figures (4)

  • Figure 1: The seven $\mathbb{F}_2$-points and the seven $\mathbb{F}_2$-lines on $\mathbb{P}^2$
  • Figure 2: $X$ of type $4$. The Galois action on the singular fibers of $X$ and the seven $\mathbb{F}_2$-points
  • Figure 3: $X$ of type $2+2$. The Galois action on the singular fibers of $\mathcal{X}_2$ over $\mathbf{k}=\mathbb{F}_2$ and the seven $\mathbb{F}_2$-points
  • Figure 4: The contraction of ${L_{45},L_{35},L_{25},L_{15}\subset X_5}$ from $X_5$ to $\mathbb{P}^2$ over $L$

Theorems & Definitions (133)

  • Proposition 1.1
  • Theorem 1.2: Counting Sarkisov links over $\mathbf{k}=\mathbb{F}_2$
  • Remark 1.3
  • Example 1.4
  • Proposition 1.5
  • Proposition 1.6: Schneider-relations
  • Theorem 1.7
  • Theorem 2.1: LamySchneider
  • Lemma 2.2: Schneider-relations
  • Lemma 2.3
  • ...and 123 more