Unitals of PG(2,q^2) containing conics

TL;DR

This paper characterizes Baker-Ebert-Hirschfeld-Szonyi unitals in PG(2,q^2) as those that are unions of q conics, providing a geometric criterion that distinguishes them from other known unitals.

Contribution

It proves that the Baker-Ebert-Hirschfeld-Szonyi unitals are uniquely characterized by being unions of q conics in PG(2,q^2).

Findings

Baker-Ebert-Hirschfeld-Szonyi unitals are unions of q conics.

This property uniquely characterizes these unitals.

The result distinguishes these from classical and other known unitals.

Abstract

A unital in PG(2,q^2) is a set U of q^3+1 points such that each line meets U in 1 or q+1 points. The well known example is the classical unital consisting of all absolute points of a non-degenerate unitary polarity of PG(2,q^2). Unitals other than the classical one also exist in PG(2,q^2) for every q>2. Actually, all known unitals are of Buekenhout-Metz type and they can be obtained by a construction due to Buekenhout. The unitals constructed by Baker-Ebert, and independently by Hirschfeld-Szonyi, are the union of q conics. Our Theorem 1.1 shows that this geometric property characterizes the Baker-Ebert-Hirschfeld-Szonyi unitals.