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Automorphisms of surfaces over fields of positive characteristic

Yifei Chen, Constantin Shramov

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.

Automorphisms of surfaces over fields of positive characteristic

Abstract

We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and -Jordan property. In particular, we show that the Cremona group of rank over a field of characteristic is -Jordan, and the birational automorphism group of an arbitrary geometrically irreducible algebraic surface is nilpotently -Jordan of class at most . 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.

Paper Structure

This paper contains 14 sections, 63 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Automorphism groups
  4. Group of connected components
  5. Minimal models
  6. Abelian varieties
  7. Varieties of non-negative Kodaira dimension
  8. Jordan property for curves
  9. Cremona group
  10. Jordan property for surfaces
  11. Surfaces of zero Kodaira dimension
  12. Fixed points in arbitrary dimension
  13. Nilpotent groups
  14. Conclusion

Key Result

Theorem 1.4

Let $n$ be a positive integer. Then there exists a constant $J(n)$ such that for every prime $p$ and every field $\Bbbk$ of characteristic $p$, every finite subgroup $G$ of $\mathop{\mathrm{GL}}\nolimits_n(\Bbbk)$ contains a normal abelian subgroup of order coprime to $p$ and index at most $J(n)\cdo

Theorems & Definitions (143)

  • Definition 1.1: see Popov
  • Definition 1.2: Hu
  • Definition 1.3: Hu, see also BrauerFeit
  • Theorem 1.4: BrauerFeit, LarsenPink
  • Theorem 1.5: Hu
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 133 more