Ovoid (projective geometry)

In PG(3,q), with q a prime power greater than 2, an ovoid is a set of q^2%2B1 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 q^2%2B1 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^{2 h%2B1} 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%2B1 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

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