Buekenhout-Metz unital

A type of unital constructed from via the construction of a translation plane (cf. Translation surface). Let be a hyperplane of and let be a spread, that is a set of lines, necessarily in number, partitioning . Define an incidence structure (cf. Incidence system), where the elements of are the points of and the lines of . The elements of are the planes of meeting in precisely a line of and the single element . Incidence is inclusion. Then is a projective plane, which is Desarguesian (cf. Desargues geometry) if is regular, that is, if it has the property that three tranversals of three lines of are transversals of lines of .

Now, let be an ovoid, that is, a set of points, no three collinear, in a hyperplane other than such that is the single point , where is not on the line . Let be the line of through and let be a point of other than . Then, with the cone with vertex and base ,

is the eponymous unital in . If is Desarguesian, both the Tits ovoid when with and a suitably chosen elliptic quadric for arbitrary with give a unital, also called in this case a Hermitian arc, that is not a Hermitian curve [a1], [a3]. An explicit equation of degree can be given [a2].

References

[a1]	F. Buekenhout, "Existence of unitals in finite translation planes of order with a kernel of order " Geom. Dedicata , 5 (1976) pp. 189–194
[a2]	J.W.P. Hirschfeld, "Finite projective spaces of three dimensions" , Oxford Univ. Press (1985)
[a3]	R. Metz, "On a class of unitals" Geom. Dedicata , 8 (1979) pp. 125–126