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A determinant on birational maps of Severi-Brauer surfaces

Elias Kurz

TL;DR

This work determines the abelianization of the birational automorphism group of a non-trivial Severi-Brauer surface over a perfect field by introducing a determinant det on Aut_k(S) and extending it to Bir via Sarkisov links between S and S^{op}. The determinant, together with a BSY-type map, yields an explicit abelianization: $Bir_k(S)^{ab} \cong (\bigoplus_{p\in \mathcal{E}_3\setminus\{q\}} \mathbb{Z}/3\mathbb{Z}) \oplus (\bigoplus_{p\in \mathcal{E}_6} \mathbb{Z}) \oplus DET$, where $DET = \det(Aut_k(S) \subseteq k^*/(k^*)^3)$; elementary relations are shown to have trivial determinant, ensuring det_R descends to a well-defined homomorphism. The results reveal that automorphisms can contribute non-trivial elements to the abelianization, and in particular, when no 6-points exist the abelianization is 3-torsion. The paper also derives maximal subgroups of $Bir_k(S)$ from these invariants and discusses connections to central simple algebras and norm maps.

Abstract

We define a determinant on the group of automorphisms of non-trivial Severi-Brauer surfaces over a perfect field. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this determinant and a group homomorphism found in arXiv:2211.17123 we can determine the abelianization of the group of birational transformations of a non-trivial Severi-Brauer surface. This is the first example of an abelianization of the group of birational transformations of a geometrically rational surface where the automorphisms are non trivial. Using the abelianization we find maximal subgroups of the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field.

A determinant on birational maps of Severi-Brauer surfaces

TL;DR

This work determines the abelianization of the birational automorphism group of a non-trivial Severi-Brauer surface over a perfect field by introducing a determinant det on Aut_k(S) and extending it to Bir via Sarkisov links between S and S^{op}. The determinant, together with a BSY-type map, yields an explicit abelianization: , where ; elementary relations are shown to have trivial determinant, ensuring det_R descends to a well-defined homomorphism. The results reveal that automorphisms can contribute non-trivial elements to the abelianization, and in particular, when no 6-points exist the abelianization is 3-torsion. The paper also derives maximal subgroups of from these invariants and discusses connections to central simple algebras and norm maps.

Abstract

We define a determinant on the group of automorphisms of non-trivial Severi-Brauer surfaces over a perfect field. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this determinant and a group homomorphism found in arXiv:2211.17123 we can determine the abelianization of the group of birational transformations of a non-trivial Severi-Brauer surface. This is the first example of an abelianization of the group of birational transformations of a geometrically rational surface where the automorphisms are non trivial. Using the abelianization we find maximal subgroups of the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field.

Paper Structure

This paper contains 10 sections, 41 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mori Fibre Spaces and Sarkisov links
  4. Properties of Severi-Brauer surfaces
  5. The determinant map
  6. Rewriting elementary relations
  7. The determinant of an elementary relation
  8. The relation with only one class of points
  9. The relation with two classes of $3$-points
  10. The abelianization of $Bir_{\textbf{k}}(S)$

Key Result

Theorem A

Let $S$ be a non-trivial Severi Brauer surface over $\textbf{k}$ and $S^{op}$ its opposite Severi-Brauer surface. Let $\chi, \chi_i: S \dashrightarrow S^{op}, \, \tau, \tau_i:S^{op} \dashrightarrow S$ be $\textbf{k}$-Sarkisov links based on $3$-points such that there are $\alpha_i, \delta_i \in Aut_

Figures (1)

  • Figure 1: Relation of Lemma \ref{['ElRel']}. The centre is a del Pezzo surface $X_3$ of degree $3$ of Picard rank $3$. Each segment denotes the blow-up of a point of degree $3$ (Figure from BSY).

Theorems & Definitions (85)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Lamy_2020
  • Theorem 2.6: Iskovskikh and Lamy_2020
  • Lemma 2.7: BSY
  • Definition 2.8
  • ...and 75 more