Buekenhout-Metz unital

A type of unital constructed from $ { \mathop{\rm PG} } ( 4,q ) $ via the construction of a translation plane (cf. Translation surface). Let $ \Pi $ be a hyperplane of $ { \mathop{\rm PG} } ( 4,q ) $ and let $ S $ be a spread, that is a set of lines, necessarily $ q ^ {2} + 1 $ in number, partitioning $ \Pi $. Define an incidence structure $ {\mathcal I} = ( {\mathcal P}, {\mathcal B} ) $( cf. Incidence system), where the elements of $ {\mathcal P} $ are the $ q ^ {4} $ points of $ { \mathop{\rm PG} } ( 4,q ) \backslash \Pi $ and the $ q ^ {2} + 1 $ lines of $ S $. The elements of $ {\mathcal B} $ are the $ q ^ {4} + q ^ {2} $ planes of $ { \mathop{\rm PG} } ( 4,q ) $ meeting $ \Pi $ in precisely a line of $ S $ and the single element $ S $. Incidence is inclusion. Then $ {\mathcal I} $ is a projective plane, which is Desarguesian (cf. Desargues geometry) if $ S $ is regular, that is, if it has the property that three tranversals of three lines of $ S $ are transversals of $ q + 1 $ lines of $ S $.

Now, let $ O $ be an ovoid, that is, a set of $ q ^ {2} + 1 $ points, no three collinear, in a hyperplane $ \Pi ^ \prime $ other than $ \Pi $ such that $ O \cap \Pi $ is the single point $ P $, where $ P $ is not on the line $ \Pi \cap \Pi ^ \prime $. Let $ {\mathcal l} $ be the line of $ S $ through $ P $ and let $ Q $ be a point of $ {\mathcal l} $ other than $ P $. Then, with $ QO $ the cone with vertex $ Q $ and base $ O $,

$$ {\mathcal U} = ( QO \backslash {\mathcal l} ) \cup \{ {\mathcal l} \} $$

is the eponymous unital in $ {\mathcal I} $. If $ {\mathcal I} $ is Desarguesian, both the Tits ovoid when $ q = 2 ^ {2e + 1 } $ with $ e \geq 1 $ and a suitably chosen elliptic quadric for arbitrary $ q $ with $ q > 2 $ give a unital, also called in this case a Hermitian arc, that is not a Hermitian curve [a1], [a3]. An explicit equation of degree $ 2q $ can be given [a2].

References

[a1]	F. Buekenhout, "Existence of unitals in finite translation planes of order $q^2$ with a kernel of order $q$" 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