Abstract This paper addresses a number of problems concerning Buekenhout-Tits unitals in $${{\,\textrm{PG}\,}}(2, q^2)$$ PG ( 2 , q 2 ) , where $$q = 2^{2e + 1}$$ q = 2 2 e + 1 and $$e \ge 1$$ e ≥ 1 . We show that all Buekenhout-Tits unitals are equivalent under $${{\,\textrm{PGL}\,}}(3, q^2)$$ PGL ( 3 , q 2 ) [addressing an open problem in Barwick and Ebert (Unitals in Projective Planes. Springer Monographs in Mathematics. Springer, New York, 2008)], explicitly describe their stabiliser in $$\textrm{P}\Gamma \textrm{L}(3, q^2)$$ P Γ L ( 3 , q 2 ) [expanding Ebert’s work in Ebert (J Algebraic Comb 6(2):133–140, 1997)], and show that lines meet the feet of points not on $$\ell _{\infty }$$ ℓ ∞ in at most four points. Finally, we show that feet of points not on $$\ell _{\infty }$$ ℓ ∞ are not always a $$\{0, 1, 2, 4\}$$ { 0 , 1 , 2 , 4 } -set, in contrast to what happens for Buekenhout-Metz unitals Abarzúa et al (Adv Geom 18(2):229–236, 2018).

Abstract If an $${\mathbb {F}}_q$$ F q -linear set $$L_U$$ L U in a projective space is defined by a vector subspace U which is linear over a proper superfield of $${\mathbb {F}}_{q}$$ F q , then all of its points have weight at least 2. It is known that the converse of this statement holds for linear sets of rank h in $$\mathrm {PG}(1,q^h)$$ PG ( 1 , q h ) but for linear sets of rank $$k k < h the converse of this statement is in general no longer true. The first part of this paper studies the relation between the weights of points and the size of a linear set, and introduces the concept of the geometric field of linearity of a linear set. This notion will allow us to show the main theorem, stating that for particular linear sets without points of weight 1, the converse of the above statement still holds as long as we take the geometric field of linearity into account.