On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals

TL;DR

This paper proves all Buekenhout-Tits unitals in PG(2,q^2) are PGL-equivalent, describes their stabilisers, and investigates properties of feet of points, revealing new geometric insights and distinctions from Buekenhout-Metz unitals.

Contribution

It establishes the PGL-equivalence of all Buekenhout-Tits unitals, explicitly describes their stabilisers, and analyzes the properties of feet of points in these unitals.

Findings

All Buekenhout-Tits unitals are PGL-equivalent.

The PΓL-stabiliser of these unitals is explicitly described.

Feet of points not on the line at infinity can have more than four points.

Abstract

This paper addresses a number of problems concerning Buekenhout-Tits unitals in $PG(2,q^2)$, where $q = 2^{e+1}$ and $e \geq 1$. We show that all Buekenhout-Tits unitals are $PGL$-equivalent (addressing an open problem in [S. Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in Mathematics. Springer, New York, 2008.]), explicitly describe their $P\Gamma L$-stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals. J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet of points no on $\ell_\infty$ in at most four points. Finally, we show that feet of points not on $\ell_\infty$ are not always a $\{0,1,2,4\}$-set, in contrast to what happens for Buekenhout-Metz unitals [N. Abarz\'ua, R. Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom. 18.2 (2018), 229-236].