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A type of [[Unital|unital]] constructed from  $  { \mathop{\rm PG} } ( 4,q ) $
via the construction of a translation plane (cf. [[Translation surface|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|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|projective plane]], which is Desarguesian (cf. [[Desargues geometry|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(2)|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 [[#References|[a1]]], [[#References|[a3]]]. An explicit equation of degree  $  2q $
can be given [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Buekenhout,   "Existence of unitals in finite translation planes of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101049.png" /> with a kernel of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101050.png" />"  ''Geom. Dedicata'' , '''5'''  (1976)  pp. 189–194</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W.P. Hirschfeld,   "Finite projective spaces of three dimensions" , Oxford Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Metz,   "On a class of unitals"  ''Geom. Dedicata'' , '''8'''  (1979)  pp. 125–126</TD></TR></table>