Projective Geometry - Lecture Notes Chapter 4
4.3 Quadratic Sets in Spaces of Small Dimension
Def: A nonempty set O of points in a projective plane is called an oval if no three points of O are collinear and each point of O is on exactly one tangent.
Theorem 4.3.1: Let Q be a quadratic set in a projective plane P. Then Q is the empty set, just one point, one line, an oval, the set of points on two lines, or the whole set of points.
Hence there is only one type of nonempty, nondegenerate quadratic sets in a projective plane, namely the ovals.
Def: Let P be a d-dimensional projective space. An ovoid is a nonempty set O of points of P satisfying :
Now let d = 3.
- No three points of O are collinear.
- For each point P on O, the tangents through P cover exactly a hyperplane.
A set K of points of P is called a cone if there are a plane , and oval O in , and a point V such that K consists of the points on the lines VX with XO. We call V the vertex of the cone K.
A hyperboloid is the set of points incident with the lines of a regulus.
Theorem 4.3.2: Let Q be a quadratic set in a 3-dimensional projective space P. Then Q is a subspace, an ovoid, a cone, a hyperboloid or the union of two hyperplanes.
In particular, the nonempty, nondegenerate quadratic sets in a 3-dimensional projective space are precisely the ovoids and the hyperboloids.
4.4 Quadratic sets in Finite Projective Spaces
In this section let P = PG(d,q) and let Q be a quadratic set in P.
Lemma 4.4.1: For a point PQ\rad(Q) we denote by a (=aP) the number of Q-lines through P. Then:
- If QP is a hyperplane then QP contains exactly aq + 1 points of Q.
- We have |Q| = 1 + qd-1 + aq; in particular, a is independent of the choice of the point PQ\rad(Q).
Theorem 4.4.2: Any nonempty, nondegenerate quadratic set in P = PG(4,q) has index 2.
- In the plane (d = 2), an oval contains no lines (a = 0), so the number of points on an oval is 1 + q. For the pair of intersecting lines, a point not equal to the point of intersection, is on exactly one line contained in the set (a = 1), so the number of points is 1 + q + q = 2q + 1.
- In 3-space (d = 3) an ovoid contains no lines (a = 0) and so has 1 + q2 points; there are 2 lines through each point of a hyperboloid (a = 2) in the hyperboloid and so there are 1 + q2 + 2q = (q+1)2 points; and through each point other than the vertex of a cone there passes one line of the cone (a = 1), so the cone contains 1 + q2 + q points.
Theorem 4.4.3: Any nonempty, nondegenerate quadratic set in P = PG(2t,q) has index t.
Theorem 4.4.4: (Witt's Theorem) The index s of a nonempty, nondegenerate quadratic set in P = PG(d,q) is either d/2 if d is even, or s = (d-1)/2 or (d+1)/2 if d is odd.