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A type of [[Unital|unital]] constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110101.png" /> via the construction of a translation plane (cf. [[Translation surface|Translation surface]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110102.png" /> be a hyperplane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110103.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110104.png" /> be a spread, that is a set of lines, necessarily <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110105.png" /> in number, partitioning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110106.png" />. Define an incidence structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110107.png" /> (cf. [[Incidence system|Incidence system]]), where the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110108.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b1110109.png" /> points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101010.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101011.png" /> lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101012.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101013.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101014.png" /> planes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101015.png" /> meeting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101016.png" /> in precisely a line of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101017.png" /> and the single element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101018.png" />. Incidence is inclusion. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101019.png" /> is a [[Projective plane|projective plane]], which is Desarguesian (cf. [[Desargues geometry|Desargues geometry]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101020.png" /> is regular, that is, if it has the property that three tranversals of three lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101021.png" /> are transversals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101022.png" /> lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101023.png" />.
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$#C+1 = 48 : ~/encyclopedia/old_files/data/B111/B.1101010 Buekenhout\ANDMetz unital
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Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101024.png" /> be an [[Ovoid(2)|ovoid]], that is, a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101025.png" /> points, no three collinear, in a hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101026.png" /> other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101028.png" /> is the single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101030.png" /> is not on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101031.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101032.png" /> be the line of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101033.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101034.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101035.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101036.png" /> other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101037.png" />. Then, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101038.png" /> the cone with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101039.png" /> and base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101040.png" />,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101041.png" /></td> </tr></table>
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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 $.
  
is the eponymous unital in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101043.png" /> is Desarguesian, both the Tits ovoid when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101045.png" /> and a suitably chosen elliptic quadric for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101046.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101047.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111010/b11101048.png" /> can be given [[#References|[a2]]].
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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====
 
====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>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  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</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>

Latest revision as of 08:48, 26 March 2023


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
How to Cite This Entry:
Buekenhout-Metz unital. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buekenhout-Metz_unital&oldid=12945
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