A geometric proof of Wilbrink's characterization of even order classical unitals

TL;DR

This paper provides a geometric proof that characterizes classical unitals of even order using conditions on O'Nan configurations and parallelism, avoiding complex group theory classifications.

Contribution

It offers a new geometric proof of Wilbrink's characterization, eliminating the need for deep group-theoretic results.

Findings

Classical unitals of even order are characterized by specific geometric conditions.

The proof avoids reliance on the classification of finite groups with split BN-pairs.

Conditions involve the absence of certain line configurations and a notion of parallelism.

Abstract

Using geometric methods and without invoking deep results from group theory, we prove that a classical unital of even order $n\geq4$ is characterized by two conditions (I) and (II): (I) is the absence of O'Nan configurations of four distinct lines intersecting in exactly six distinct points; (II) is a notion of parallelism. This was previously proven by Wilbrink (1983), where the proof depends on the classification of finite groups with a split BN-pair of rank 1.