Ovoid (projective geometry)

In PG(3,q), with q a prime power greater than 2, an ovoid is a set of q² + 1 points, no three of which collinear (the maximum size of such a set).^[1] When q = 2 the largest set of non-collinear points has size eight and is the complement of a plane.^[2]

An important example of an ovoid in any finite projective three-dimensional space are the q² + 1 points of an elliptic quadric (all of which are projectively equivalent).

When q is odd or q = 4, no ovoids exist other than the elliptic quadrics.^[3]

When q = 2^{2h + 1} another type of ovoid can be constructed : the Tits ovoid, also known as the Suzuki ovoid. It is conjectured that no other ovoids exist in PG(3,q).

Through every point P on the ovoid, there are exactly q + 1 tangents, and it can be proven that these lines are exactly the lines through P in one specific plane through P. This means that through every point P in the ovoid, there is a unique plane intersecting the ovoid in exactly one point.^[4] Also, if q is odd or q = 4 every plane which is not a tangent plane meets the ovoid in a conic.^[5]

See also

• Ovoid (polar space)
• Oval (projective plane)
• Inversive plane

Notes

1. ^ more properly the term should be ovaloid and ovoid has a different definition which extends to projective spaces of higher dimension. However, in dimension 3 the two concepts are equivalent and the ovoid terminology is almost universally used, except most notably, in Hirschfeld.
2. ^ Hirschfeld 1985, pg.33, Theorem 16.1.3
3. ^ Barlotti 1955 and Panella 1955
4. ^ Hirschfeld 1985, pg. 34, Lemma 16.1.6
5. ^ Hirschfeld 1985, pg.35, Corollary

References

• Barlotti, A. (1955), "Un' estensione del teorema di Segre-Kustaanheimo", Boll. Un. Mat. Ital. 10: 96-98 
• Hirschfeld, J.W.P. (1985), Finite Projective Spaces of Three Dimensions, New York: Oxford University Press, ISBN 0-19-853536-8 
• Panella, G. (1955), "Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito", Boll. Un. Mat. Ital. 10: 507-513