Projective Geometry - Lecture Notes Chapter 4

4.5 Elliptic, Parabolic and Hyperbolic Quadratic Sets

Def: Let Q be a nondegenerate quadratic set in a d-dimensional projective space P. If d is even and the index of Q is d/2 then Q is called parabolic. If d is odd and the index of Q is d-1/2 then Q is called elliptic. Finally, if d is odd and the index of Q is d+1/2 then Q is called hyperbolic.

Examples:

  1. Theorem 4.4.4 can be reformulated as: Any nonempty nondegenerate quadratic set of a finite projective space is elliptic, parabolic or hyperbolic.
  2. The parabolic quadratic sets of a projective plane are the ovals.
  3. In a 3-dimensional projective space, the elliptic quadratic sets are the ovoids and the hyperbolic quadratic sets are the hyperboloids.
We now give a general definition of a cone.

Def: Let H be a hyperplane of a projective space P, and let V be a point outside of H. If Q* is a nondegenerate quadratic set of H the quadratic set

Q = union (VX), with X in Q*
is called a cone with vertex V over Q*.

Parabolic Quadratic Sets

Theorem 4.5.1: Let Q be a parabolic quadratic set in a 2t-dimensional projective space P with t > 1.
  1. Let H = QP be a tangent hyperplane. Then Q' = Q intersect H is a cone over a parabolic quadratic set with vertex P.
  2. Let H* be a hyperplane that is not a tangent hyperplane. Then Q* = Q intersect H* is an elliptic or hyperbolic quadratic set.

Corollary 4.5.2: Let Q be a nonempty, nondegenerate quadratic set in P = PG(4,q). Then Q induces in any tangent hyperplane a cone, and in any other hyperplane an ovoid or a hyperboloid. Furthermore, Q consists of exactly q3 + q2 + q + 1 points, the number of hyperplanes in which Q induces an ovoid is q2(q2-1)/2, and the number of hyperplanes in which Q induces a hyperboloid is q2(q2+1)/2.

Hyperbolic Quadratic Sets

Theorem 4.5.3: Let Q be a hyperbolic quadratic set of a (2t+1)-dimensional projective space P with t > 1.
  1. If H = QP is a tangent hyperplane then Q' = Q intersect H is a cone over a hyperbolic quadratic set with vertex P.
  2. If H* is a hyperplane which is not a tangent hyperplane then Q* = Q intersect H* is a parabolic quadratic set.
Theorem 4.5.4: Let Q be a hyperbolic quadratic set in P = PG(5,q). Then a = (q+1)2 and
|Q| = q4 + q3 + 2q2 + q + 1 = (q2 + q + 1)(q2 + 1).

Def: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. We say that two Q-planes pi1 and pi2 are equivalent (written pi1 ~ pi2) if pi1 and pi2 are equal or intersect in precisely one point.

Lemma 4.5.5: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. Then the relation ~ is an equivalence relation.

Theorem 4.5.6: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. Then the set of all Q-planes is partitioned into exactly two equivalence classes with respect to ~.

Def: A generalized quadrangle is a rank 2 geometry consisting of points and lines such that

  1. Any two distinct points are on at most one line.
  2. All lines are incident with the same number of points; all points are incident with the same number of lines.
  3. If P is a point outside a line g, then there is precisely one line through P intersecting g.

Theorem 4.5.8: The geometry consisting of the points, lines and planes of a hyperbolic quadratic set of a 5-dimensional projective space has the following diagram.

o ----- o ====== o