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Groups acting on cubic surfaces in characteristic zero

Jonathan M. Smith

Abstract

For every field $k$ of characteristic zero, we determine the groups that act as automorphisms on a smooth cubic surface over $k$. We also determine the groups that act on $k$-rational, stably $k$-rational, or $k$-unirational smooth cubic surfaces.

Groups acting on cubic surfaces in characteristic zero

Abstract

For every field of characteristic zero, we determine the groups that act as automorphisms on a smooth cubic surface over . We also determine the groups that act on -rational, stably -rational, or -unirational smooth cubic surfaces.
Paper Structure (10 sections, 14 theorems, 32 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group theoretic notation
  4. Del Pezzo surfaces
  5. Automorphisms over algebraically closed fields
  6. The Classification
  7. Subgroups of 5A, 8A, and 12A
  8. Subgroups of 3C
  9. Proof of Theorem \ref{['thm:main']}
  10. Rationality

Key Result

Theorem 1.1

Let $k$ be a field of characteristic zero. If $G$ is a maximal group in $\mathcal{P}_{3,k}$, then $G$ is one of the groups in the table. Each group appears in $\mathcal{P}_{3,k}$ if and only if the condition in the third column is satisfied. When the condition in the third column is satisfied, the f Moreover, each group in the table is maximal in $\mathcal{P}_{3,k}$ for some choice of $k$.

Figures (1)

  • Figure 1: Specialization of strata in $\mathcal{M}_{\text{cub}}$ for $k = \bar{k}$ with $\text{char}(k) = 0$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 27 more