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Automorphisms of del Pezzo surfaces in characteristic 2

Igor Dolgachev, Gebhard Martin

TL;DR

This work completes the classification of automorphism groups for del Pezzo surfaces of degrees 1 and 2 over algebraically closed fields of characteristic 2, finishing the overarching automorphism program across all characteristics. Central to the analysis are anti-canonical/anti-bicanonical maps, normal forms, and the geometric features that survive in characteristic 2, such as fake bitangents, odd theta characteristics, and strange genus-one fibrations. The authors derive explicit normal forms for degree 2 and 1 surfaces, relate automorphism groups to Weyl groups W(E_7) and W(E_8), and provide a complete enumeration of possible Aut(X) up to conjugacy, including a maximal order of 1920 for degree 1. They also connect the characteristic-2 cases to characteristic-0 lifts where possible and detail the conjugacy-class data in the corresponding Weyl groups, enriching the Cremona-birational classification landscape with characteristic-2 phenomena.

Abstract

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

Automorphisms of del Pezzo surfaces in characteristic 2

TL;DR

This work completes the classification of automorphism groups for del Pezzo surfaces of degrees 1 and 2 over algebraically closed fields of characteristic 2, finishing the overarching automorphism program across all characteristics. Central to the analysis are anti-canonical/anti-bicanonical maps, normal forms, and the geometric features that survive in characteristic 2, such as fake bitangents, odd theta characteristics, and strange genus-one fibrations. The authors derive explicit normal forms for degree 2 and 1 surfaces, relate automorphism groups to Weyl groups W(E_7) and W(E_8), and provide a complete enumeration of possible Aut(X) up to conjugacy, including a maximal order of 1920 for degree 1. They also connect the characteristic-2 cases to characteristic-0 lifts where possible and detail the conjugacy-class data in the corresponding Weyl groups, enriching the Cremona-birational classification landscape with characteristic-2 phenomena.

Abstract

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.
Paper Structure (23 sections, 30 theorems, 83 equations, 4 tables)

This paper contains 23 sections, 30 theorems, 83 equations, 4 tables.

Table of Contents

  1. Notation
  2. Del Pezzo surfaces of degree $\ge 3$
  3. Degree $\ge 5$
  4. Quartic del Pezzo surfaces
  5. Cubic surfaces
  6. Del Pezzo surfaces of degree $2$
  7. The anti-canonical map
  8. Normal forms
  9. Fake bitangents and odd theta characteristics
  10. Strange elliptic fibrations
  11. Automorphism groups of del Pezzo surfaces of degree $2$
  12. Preliminaries
  13. Classification
  14. Conjugacy classes and comparison with the classification in characteristic $0$
  15. Del Pezzo surfaces of degree $1$
  16. ...and 8 more sections

Key Result

Theorem 1

A finite group $G$ is realized as the automorphism group $\Aut(X)$ of a del Pezzo surface $X$ of degree $2$ (resp. $1$) over an algebraically closed field $k$ of characteristic ${\rm char}(k)= 2$ if and only if $G$ is listed in Table tbl:autodp2 (resp. Table tbl:autodp1) in the Appendix.

Theorems & Definitions (61)

  • Theorem
  • proposition 1
  • proof
  • proposition 2
  • remark 1
  • theorem 1
  • proof
  • lemma 1
  • proof
  • lemma 2
  • ...and 51 more