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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 ![]() ![]() |
[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 |
Buekenhout-Metz unital. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buekenhout-Metz_unital&oldid=12945