A combinatorial characterization of classical unitals
In 1992, Jef Thas published the following beautiful theorem [see J. Algebr. Comb. 1, No. 1, 97-102 (1992; Zbl 0784.51023)]: if for a unital \(U\) in a finite Desarguesian projective plane the intersection points of tangents through any point outside \(U\) are collinear, then \(U\) is the classical Hermitian unital. In the present paper, the authors weaken this condition by restricting to points lying on either of two fixed tangent lines. The techniques boil down to saving (and modifying) what they can from Thas' proof, and using a recent powerful characterization of elliptic quadrics in \(\text{PG}(3,q)\) due to Matthew Brown, stating that an ovoid in \(\text{PG} (3,q)\) is an elliptic quadric whenever at least one plane section is a conic [\textit{M. Brown}, J. Lond. Math. Soc., II. Ser. 62, No. 2, 569-582 (2000; Zbl 1038.51008)]. It may sound trivial, but it is well done, and the result is a very nice improvement of the one of Thas mentioned above.
scientific article; zbMATH DE number 1742833