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 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
How to Cite This Entry:
Buekenhout-Metz unital. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buekenhout-Metz_unital&oldid=46172
This article was adapted from an original article by J.W.P. Hirschfeld (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article