Eggs in finite projective spaces and unitals in translation planes

Abstract

Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the André/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of \(\textrm{PG}(4m-1,q)\), \(m\ge 1\), 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 \(\textrm{PG}(20,3)\) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in \(\textrm{PG}(19,3)\). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.