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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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| | + | $#A+1 = 48 n = 2 |
| | + | $#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>
| + | 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]]]. | + | 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> |
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 |