Eggs in finite projective spaces and unitals in translation planes

TL;DR

This paper explores the relationship between eggs and unitals in higher-dimensional translation planes, leading to the construction of a potentially new unital in a Dickson semifield plane of order 3^{10}.

Contribution

It establishes a novel connection between eggs and unitals in higher dimensions and constructs a new unital in a Dickson semifield plane.

Findings

Constructed a unital in the Dickson semifield plane of order 3^{10}.

The unital is not polar, indicating novelty.

Linked eggs and unitals in higher-dimensional translation planes.

Abstract

Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the Andr\'e/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of PG(4m-1,q), m>=1, are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order 3^{10}, which is represented in PG(20,3) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in PG(19,3). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.