# 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 |

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Buekenhout-Metz unital.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Buekenhout-Metz_unital&oldid=46172