A Generalization of von Staudt's Theorem on Cross-Ratios

TL;DR

This paper generalizes von Staudt's theorem, showing that permutations preserving harmonic quadruples are projective semilinear maps, and explores the transitivity properties of supergroups over algebraically closed fields.

Contribution

It extends von Staudt's theorem to broader contexts and characterizes supergroups of projective semilinear groups in algebraic geometry.

Findings

Permutations preserving harmonic quadruples are projective semilinear maps.

Supergroups of the projective semilinear group are 4-transitive over certain fields.

The generalization applies to algebraically closed fields with transcendence degree at least 1.

Abstract

A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.