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Umemura Quadric Fibrations and Maximal Subgroups of $Cr_n(\mathbb{C})$

Enrica Floris, Sokratis Zikas

TL;DR

This work advances the understanding of maximal connected algebraic subgroups of the Cremona group Cr_n(k) by focusing on Umemura quadric fibrations Q_g, which provide a family of highly symmetric rational varieties. The authors develop a precise equivariant birational framework, including minimal G-equivariant log resolutions and detailed analyses of automorphism groups, Mori cones, and orbit structures, to characterize when Aut^\circ(Q_g) is maximal in Cr_n(k). A key insight is that the maximality is governed by the square-free part h of g, with Aut^\circ(Q_g) maximal precisely when h is constant or has at least four distinct roots, and conjugacy is controlled by PGL_2(k) action on h. They further show that, in the case of exactly two roots, Aut^\circ(Q_g) sits inside a unique maximal subgroup PSO_{n+1}(k), while for more roots Aut^\circ(Q_g) equals SO_n(k); these results yield infinite families of pairwise non-conjugate maximal connected subgroups in higher-dimensional Cremona groups. The methods combine MMP, Sarkisov theory, and explicit equivariant resolutions to connect automorphism groups to birational Mori fibre spaces, with potential impact on the broader classification program for maximal subgroups in Cr_n(k).

Abstract

We study the equivariant geometry of special quadric fibrations, called Umemura quadric fibrations, as well as the maximality of their automorphism groups inside $Cr_n(\mathbb{C})$. We produce infinite families of pairwise non-conjugate maximal connected algebraic subgroups of $Cr_n(\mathbb{C})$.

Umemura Quadric Fibrations and Maximal Subgroups of $Cr_n(\mathbb{C})$

TL;DR

This work advances the understanding of maximal connected algebraic subgroups of the Cremona group Cr_n(k) by focusing on Umemura quadric fibrations Q_g, which provide a family of highly symmetric rational varieties. The authors develop a precise equivariant birational framework, including minimal G-equivariant log resolutions and detailed analyses of automorphism groups, Mori cones, and orbit structures, to characterize when Aut^\circ(Q_g) is maximal in Cr_n(k). A key insight is that the maximality is governed by the square-free part h of g, with Aut^\circ(Q_g) maximal precisely when h is constant or has at least four distinct roots, and conjugacy is controlled by PGL_2(k) action on h. They further show that, in the case of exactly two roots, Aut^\circ(Q_g) sits inside a unique maximal subgroup PSO_{n+1}(k), while for more roots Aut^\circ(Q_g) equals SO_n(k); these results yield infinite families of pairwise non-conjugate maximal connected subgroups in higher-dimensional Cremona groups. The methods combine MMP, Sarkisov theory, and explicit equivariant resolutions to connect automorphism groups to birational Mori fibre spaces, with potential impact on the broader classification program for maximal subgroups in Cr_n(k).

Abstract

We study the equivariant geometry of special quadric fibrations, called Umemura quadric fibrations, as well as the maximality of their automorphism groups inside . We produce infinite families of pairwise non-conjugate maximal connected algebraic subgroups of .
Paper Structure (12 sections, 25 theorems, 55 equations)

This paper contains 12 sections, 25 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Extremal Divisorial Contractions
  4. Action on a product of quadrics
  5. A useful lemma
  6. Umemura quadric fibrations
  7. Definition and basic properties
  8. Automorphism group
  9. A structural result for equivariant birational maps to $\mathcal{Q}_g$.
  10. An equivariant resolution
  11. Equivariant Geometry of $\mathcal{Q}_g$
  12. Maximality of $\mathrm{Aut}^\circ(\mathcal{Q}_g)$

Key Result

Theorem 1

Let $g \in \mathbb \textbf{k}[t_0, t_1]$ be a homogeneous polynomial of degree $2a$ and $\pi\colon \mathcal{Q}_g \to\mathbb{P}^1$ the associated Umemura quadric fibration. Write $g = f^2h$, where $f, h \in \textbf{k}[t_0,t_1]$ are homogeneous polynomials with $h$ being square-free. Then $\mathrm{Aut

Theorems & Definitions (58)

  • Theorem 1: = Theorem \ref{['thm:mainTheorem']}
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 48 more