Unitals with many Baer secants through a fixed point

TL;DR

This paper characterizes certain unitals in projective planes as ovoidal Buekenhout-Metz unitals based on the number of Baer secants through a fixed point, providing a near-complete classification under these conditions.

Contribution

It establishes a new characterization of ovoidal Buekenhout-Metz unitals using the number of Baer secants through a fixed point in PG(2,q^2).

Findings

Unitals with many Baer secants are ovoidal Buekenhout-Metz unitals.

The result applies for both even and odd q with specific bounds.

Provides a near-complete classification based on secant line properties.

Abstract

We show that a unital $U$ in $\mathrm{PG}(2,q^2)$ containing a point $P$, such that at least $q^2-\epsilon$ of the secant lines through $P$ intersect $U$ in a Baer subline, is an ovoidal Buekenhout-Metz unital (where $\epsilon\approx 2q$ for $q$ even and $\epsilon\approx q^{3/2}/2$ for $q$ odd).