View source for Ovoid

==Ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201001.png" />-dimensional projective space.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201002.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201003.png" />-dimensional [[Projective space|projective space]] over the [[Galois field|Galois field]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201004.png" />. An ovoid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201006.png" /> is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201007.png" /> points, no three of which are collinear. (Note that in [[#References|[a2]]] and some other older publications, an ovoid is called an ovaloid.) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201008.png" />, then an ovoid is a maximum-sized set of points, no three collinear, but in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o1201009.png" /> the complement of a plane is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010010.png" /> points, no three collinear. More information about ovoids can be found in [[#References|[a2]]] and [[#References|[a5]]]. The survey paper [[#References|[a4]]] gives further details, especially regarding more recent work, with the exception of the recent (1998) result in [[#References|[a1]]].

The only known ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010011.png" /> (as of 1998) are the elliptic quadrics (cf. also [[Quadric|Quadric]]), which exist for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010012.png" />, and the Tits ovoids, which exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010013.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010014.png" /> is odd. There is a single orbit of elliptic quadrics under the homography group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010015.png" />, and one can take as a representative the set of points

<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/o/o120/o120100/o12010016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010017.png" /> is irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010018.png" />. The stabilizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010019.png" /> of an elliptic quadric is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010020.png" />, acting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010021.png" />-transitively on its points. There is a single orbit of Tits ovoids under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010022.png" />, a representative of which is given by

<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/o/o120/o120100/o12010023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010024.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010025.png" />. The stabilizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010026.png" /> of a Tits ovoid is the [[Suzuki group|Suzuki group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010027.png" /> acting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010028.png" />-transitively on its points. A plane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010029.png" /> meets an ovoid in either a single point or in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010030.png" /> points of an oval. It is worth noting that in the case of an elliptic quadric, this oval is a conic while in the case of a Tits ovoid it is a translation oval with associated automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010031.png" />.

The classification of ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010032.png" /> is of great interest, particularly in view of the number of related structures, such as Möbius planes, symplectic polarities, linear complexes, generalized quadrangles, unitals, maximal arcs, translation planes, and ovals.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010033.png" /> is odd, then every ovoid is an elliptic quadric, but the classification problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010034.png" /> even has been resolved only (as of 1998) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010035.png" />, with the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010036.png" /> involving some computer work. These classifications rely on the classification of ovals in the projective planes over fields of order up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010037.png" />. On the other hand, there are several characterization theorems known, normally in terms of an assumption on the nature of the plane sections. One of the strongest results in this direction states that an ovoid with at least one conic among its plane sections must be an elliptic quadric. Similarly, it is known that an ovoid which admits a pencil of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010038.png" /> translation ovals is either an elliptic quadric or a Tits ovoid.

==Ovoids in generalized polygons.==
An ovoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010039.png" /> in a generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010040.png" />-gon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010041.png" /> (cf. also [[Polygon|Polygon]]) is a set of mutually opposite points (hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010042.png" /> is even) such that every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010044.png" /> is at distance at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010045.png" /> from at least one element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010046.png" />. One connection between ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010047.png" /> and ovoids in generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010048.png" />-gons is that just as polarities of projective spaces sometimes give rise to ovoids, polarities of generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010049.png" />-gons produce ovoids. Further, every ovoid of the classical symplectic generalized quadrangle (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010051.png" />-gon; cf. also [[Quadrangle|Quadrangle]]) usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010053.png" /> even, is an ovoid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010054.png" /> and conversely. It is worth noting in this context that a Tits ovoid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010055.png" /> is the set of all absolute points of a polarity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010058.png" /> odd. There are many ovoids known (1998) for most classes of generalized quadrangles, with useful characterization theorems. For the details and a survey of existence and characterization results for ovoids in generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010060.png" />-gons, see [[#References|[a6]]] and especially [[#References|[a7]]].

==Ovoids in finite classical polar spaces.==
An ovoid $\mathcal{O}$ in a finite classical [[Polar space|polar space]] $\mathcal{S}$ of rank $r\geq 2$ is a set of points of $\mathcal{S}$ which has exactly one point in common with every generator of $\mathcal{S}$. Again, there are connections between ovoids in polar spaces and ovoids in generalized $n$-gons. For example, if $H(q)$ denotes the classical generalized hexagon ($6$-gon) of order $q$ embedded on the non-singular quadric $Q(6,q)$, then a set of points $\mathcal{O}$ is an ovoid of $H(q)$ if and only if it is an ovoid of the classical polar space $Q(6,Q)$. The existence problem for ovoids has been settled for most finite classical polar spaces, see [[#References|[a3]]] or [[#References|[a5]]] for a survey which includes a list of the open cases (as of 1998).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.R. Brown,   "Ovoids of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010076.png" /> even, with a conic section"  ''J. London Math. Soc.''  (to appear)</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">  J.W.P. Hirschfeld,   J.A. Thas,   "General Galois geometries" , Oxford Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C.M. O'Keefe,   "Ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120100/o12010077.png" />: a survey"  ''Discrete Math.'' , '''151'''  (1996)  pp. 175–188</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.A. Thas,   "Projective geometry over a finite field"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry, Buildings and Foundations'' , Elsevier  (1995)  pp. Chap. 7; 295–348</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.A. Thas,   "Generalized Polygons"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry, Buildings and Foundations'' , Elsevier  (1995)  pp. Chap. 9; 295–348</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Van Maldeghem,   "Generalized polygons" , Birkhäuser  (1998)</TD></TR></table>