Embeddings of Ree unitals in a projective plane over a field

TL;DR

This paper characterizes when Ree unitals can be embedded in projective planes over fields, showing that such embeddings are unique and occur only under specific conditions involving the field and the parameter q.

Contribution

It provides a complete characterization of embeddings of Ree unitals in projective planes, linking algebraic properties of fields to geometric embeddings, and uses group classification techniques.

Findings

Embedding exists only if q=3 and the field contains _8.

Embedding is unique up to projective transformations.

The proof employs classification of maximal subgroups of Ree groups.

Abstract

We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.