Unitals in $PG(2,q^2)$ with a large 2-point stabiliser

TL;DR

This paper characterizes classical unitals in $ ext{PG}(2,q^2)$ by the size of their 2-point stabilizer subgroup, showing a large stabilizer implies the unital is classical.

Contribution

It establishes a new criterion for classicality of unitals based on the order of a 2-point stabilizer subgroup in the projective plane.

Findings

A unital is classical if and only if it has two points with a stabilizer of order $q^2-1$.

The stabilizer subgroup of size $q^2-1$ characterizes classical unitals.

Provides a group-theoretic condition for identifying classical unitals.

Abstract

Let $\cU$ be a unital embedded in the Desarguesian projective plane $\PG(2,q^2)$. Write $M$ for the subgroup of $\PGL(3,q^2)$ which preserves $\cU$. We show that $\cU$ is classical if and only if $\cU$ has two distinct points $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.