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On p-Jordan Constant of Cremona Group of Rank 2 in Odd Characteristic

Yifei Chen, Constantin Shramov

TL;DR

This work establishes a precise p-Jordan-type bound for the Cremona group Cr$_2$ over fields of odd characteristic, showing that any finite subgroup has a normal abelian subgroup of order coprime to p with index bounded by a constant times the cube of the p-Sylow subgroup size: [G:A] ≤ J_p(Cr$_2$)·|G$_{(p)}$|^3. The authors derive explicit p-dependent constants by reducing to actions on conic bundles and Del Pezzo surfaces, employing detailed analyses of projective lines and planes, and leveraging classifications of finite subgroups of PSL$_2$ and PGL$_2$, along with Serre lifting and Weyl-group techniques. The paper provides a comprehensive framework combining group-theoretic lemmas (Chermak–Delgado, extension and product results) with geometric classifications (Mitchell’s list, conic bundles, del Pezzo degrees) to obtain sharp bounds: J_p(Cr$_2$)=7200 for p≥7, 168 for p=5, and 10 for p=3, and demonstrates sharpness in the algebraically closed setting. The results extend Jordan-type insights to Cremona groups in positive characteristic and highlight the role of geometry (conic bundles and Del Pezzo surfaces) in bounding abelian normal subgroups, while also outlining the open case in characteristic 2.

Abstract

We bound the indices of normal abelian subgroups in finite groups contained in the Cremona group of rank 2 over a field of odd characteristic.

On p-Jordan Constant of Cremona Group of Rank 2 in Odd Characteristic

TL;DR

This work establishes a precise p-Jordan-type bound for the Cremona group Cr over fields of odd characteristic, showing that any finite subgroup has a normal abelian subgroup of order coprime to p with index bounded by a constant times the cube of the p-Sylow subgroup size: [G:A] ≤ J_p(Cr)·|G|^3. The authors derive explicit p-dependent constants by reducing to actions on conic bundles and Del Pezzo surfaces, employing detailed analyses of projective lines and planes, and leveraging classifications of finite subgroups of PSL and PGL, along with Serre lifting and Weyl-group techniques. The paper provides a comprehensive framework combining group-theoretic lemmas (Chermak–Delgado, extension and product results) with geometric classifications (Mitchell’s list, conic bundles, del Pezzo degrees) to obtain sharp bounds: J_p(Cr)=7200 for p≥7, 168 for p=5, and 10 for p=3, and demonstrates sharpness in the algebraically closed setting. The results extend Jordan-type insights to Cremona groups in positive characteristic and highlight the role of geometry (conic bundles and Del Pezzo surfaces) in bounding abelian normal subgroups, while also outlining the open case in characteristic 2.

Abstract

We bound the indices of normal abelian subgroups in finite groups contained in the Cremona group of rank 2 over a field of odd characteristic.

Paper Structure

This paper contains 11 sections, 53 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Preliminaries
  4. Projective linear groups
  5. Semi-direct products
  6. Extensions
  7. Projective line
  8. Projective plane
  9. Conic bundles
  10. Del Pezzo surfaces
  11. Proof of the main result

Key Result

Theorem 1.1

Let $\Bbbk$ be a field of characteristic zero, and let $n$ be a positive integer. Then there exists a constant $J(n)$ such that every finite subgroup of the group $\mathop{\mathrm{GL}}\nolimits_n(\Bbbk)$ contains a normal abelian subgroup of index at most $J(n)$.

Theorems & Definitions (106)

  • Theorem 1.1: see e.g. Jordan or Serre-FiniteGroups
  • Definition 1.2: see Popov
  • Definition 1.3: Hu
  • Theorem 1.4: see LarsenPink, cf. BrauerFeit, Weisfeiler, BajpaiDona
  • Theorem 1.5: Hu
  • Theorem 1.6: ChenShramov
  • Theorem 1.7: Yasinsky
  • Theorem 1.8: see ProkhorovShramov and Vikulova
  • Theorem 1.9
  • Example 2.1
  • ...and 96 more