[Paper Review] Automorphisms of surfaces over fields of positive characteristic

Yifei Chen, Constantin Shramov|arXiv (Cornell University)|Jun 30, 2021

Algebraic Geometry and Number Theory41 references10 citations

TL;DR

This paper establishes that the birational automorphism group of any geometrically irreducible algebraic surface over a field of positive characteristic p is nilpotently p-Jordan of class at most 2, and the Cremona group of rank 2 over such fields is p-Jordan. These results extend Jordan-type finiteness properties to positive characteristic geometry, resolving key cases of a conjectural framework for finite group actions on surfaces in this setting.

ABSTRACT

We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and $p$-Jordan property. In particular, we show that the Cremona group of rank $2$ over a field of characteristic $p>0$ is $p$-Jordan, and the birational automorphism group of an arbitrary geometrically irreducible algebraic surface is nilpotently $p$-Jordan of class at most $2$. Also, we show that the automorphism group of a smooth geometrically irreducible projective variety of non-negative Kodaira dimension is Jordan in the usual sense.

Motivation & Objective

• To extend Jordan-type finiteness properties—specifically p-Jordan and nilpotent p-Jordan properties—to automorphism and birational automorphism groups of algebraic surfaces over fields of positive characteristic.
• To address the failure of classical Jordan property in positive characteristic, particularly due to the presence of large finite simple groups like PSL₂(𝔽_{p^k}) in PGL₂(𝑘̄).
• To generalize results known in characteristic zero to positive characteristic, especially for surfaces, by introducing and proving new structural constraints on finite subgroups of birational and automorphism groups.

Proposed method

• Adapting techniques from characteristic zero, particularly those involving minimal models and fixed-point theory, to the positive characteristic setting.
• Using the theory of minimal models and the classification of surfaces to reduce the study of automorphism groups to known cases, especially del Pezzo surfaces and surfaces of non-negative Kodaira dimension.
• Applying the p-Jordan property framework via Sylow subgroup analysis and the structure of finite subgroups with order coprime to p.
• Leveraging results on linear algebraic groups and their p-Jordan properties, particularly the bound from [BrF66] and [LP11], to extend them to birational automorphism groups.
• Establishing nilpotent p-Jordan structure by analyzing the derived series and the structure of normal abelian subgroups in finite subgroups.
• Using the theory of fixed points in arbitrary dimension to control the action of finite groups on surfaces, especially in the context of minimal models.

Experimental results

Research questions

• RQ1 Is the birational automorphism group of a geometrically irreducible surface over a field of positive characteristic p nilpotently p-Jordan of class at most 2?
• RQ2 Does the Cremona group of rank 2 over a field of positive characteristic p satisfy the p-Jordan property?
• RQ3 Can the p-Jordan property be extended to automorphism groups of complete or quasi-projective varieties in positive characteristic?
• RQ4 What are the precise values of the constants e(Γ) and J(Γ) in the p-Jordan definition for automorphism and birational automorphism groups of surfaces in positive characteristic?
• RQ5 Do multiplicative bounds for indices of normal abelian subgroups exist in the context of p-Jordan groups of geometric origin, or do they fail as in the case of PGL₂(𝑘̄)?

Key findings

• The Cremona group of rank 2 over a field of positive characteristic p is p-Jordan, meaning every finite subgroup contains a normal abelian subgroup of order coprime to p and index bounded by J·|G_p|^e for constants J and e depending only on the group.
• The birational automorphism group of any geometrically irreducible algebraic surface over a field of positive characteristic p is nilpotently p-Jordan of class at most 2, indicating a strong structural constraint on finite subgroups.
• The automorphism group of a smooth geometrically irreducible projective variety of non-negative Kodaira dimension over a field of positive characteristic p is Jordan in the classical sense, extending known results from characteristic zero.
• For surfaces of zero Kodaira dimension, the p-Jordan property holds, and the structure of finite subgroups is controlled via minimal model theory and fixed-point analysis.
• The automorphism group of any projective variety over a field of positive characteristic p is p-Jordan, generalizing a result of F. Hu to the full automorphism group setting.
• The paper shows that multiplicative bounds for indices of normal abelian subgroups do not exist in general for p-Jordan groups of geometric origin, as demonstrated by the unbounded index of trivial normal abelian subgroups in PSL₂(𝔽_{p^k}) subgroups of PGL₂(𝑘̄).