Projective Geometry - Lecture Notes Chapter 4

4.1 Fundamental Definitions

 Def: Let Q be a set of points of the projective space P. A line g is a tangent of Q if either g has just one point in common with Q or each point of g is contained in Q.

For each point P of Q let the set QP consist of the point P and all the points XP such that the line XP is a tangent of Q. One calls QP the tangent space of Q at P.

We call the set Q a quadratic set of P if it satisfies:

  1. If-three-then-all axiom. Any line g that contains at least three points of Q is totally contained in Q. In other words, any line not contained in Q can meet Q in at most 2 points.
  2. Tangent-space axiom. For any point P of Q, its tangent space QP is either a hyperplane or all of P.
Examples: The empty set and any subspace of P is a quadratic set. The union of two planes is a quadratic set. In 3-space, hyperbolic quadrics and cones over a quadratic set are quadratic sets. In Euclidean geometry, spheres in 3 dimensions and conics in the plane are quadratic sets.

Lemma 4.1.1: Let Q be a quadratic set of P, and let U be a subspace of P. Then the set Q' := QU of the points of Q in U is a quadratic set of U. Moreover, we have

QP' = QPU
for all points P in Q'.

Def: Let Q be a quadratic set of P. The radical of Q is the set rad(Q) of all points P in Q with the property that QP = P.

We say that Q is nondegenerate if rad(Q) = , that is, if for each point P of Q, QP is a hyperplane.

Example: A sphere in Euclidean space is a nondegenerate quadratic set, while a cone is degenerate since its radical consists of the vertex of the cone.

Theorem 4.1.2: Let Q be a quadratic set of P.

  1. The radical of Q is a (linear) subspace of P.
  2. Let U be a complement of rad(Q) (that is, a subspace U such that Urad(Q) = and <U, rad(Q)> = P). Then Q' := QU is a nondegenerate quadratic set of U.
  3. Q can be described as follows: Q consists of all points that lie on lines that join a point of rad(Q) with a point of Q' = QU.
Remark: In view of this theorem, we can restrict ourselves to the study of nondegenerate quadratic sets.

Lemma 4.1.3: Let Q be a nondegenerate quadratic set of P. For any two distinct points P and R in Q, we have QPQR. In other words, the quadratic set that is induced by Q in a tangent space QP has a radical that consists of just one point, namely P.

Lemma 4.1.4: Let Q be a nondegenerate quadratic set of P.

  1. If P is a point of Q and W is a complement of P in QP, then Q' := QW is a nondegenerate quadratic set of W.
  2. If H is a hyperplane that is not a tangent hyperplane (tangent space) then Q' := QH is a nondegenerate quadratic set of H.

4.2 The Index of a Quadratic Set

 Def: Given a quadratic set Q, any subspace contained in Q is called a Q-subspace.

Let t-1 be the maximum dimension of a Q-subspace of a quadratic set Q. Then the integer t is called the index of Q. The Q-subspaces of dimension t-1 are also called maximal Q-subspaces.

Examples: A cone and a hyperboloid in 3-dimensional Euclidean space have index 2 since they contain lines but no planes. The quadratic set consisting of the union of two planes in a projective space has index 3. Any quadratic set which does not contain a line has index 1.

Lemma 4.2.1: Let Q be a quadratic set of index t in P. Then each point of Q is on a maximal q-subspace. More precisely: if P is a point of Q outside a (t-1)-dimensional Q-subspace U, then there is a (t-1)-dimensional Q-subspace U' through P which intersects U in a (t-2)-dimensional subspace.

Remark: An important case of the above lemma is when t = 2: Through each point of Q not on a Q-line g, there passes a Q-line which intersects g.

Lemma 4.2.2: Let Q be a quadratic set in P. Let S be a subset of Q with the property that the line through any two points of S is a Q-line. Then <S> is a Q-subspace.

Theorem 4.2.3: Let Q be a quadratic set in a d-dimensional projective space P, and let U be a maximal Q-subspace. If Q is nondegenerate, then there is a maximal Q-subspace that is skew to U.

Theorem 4.2.4: Let Q be a nondegenerate quadratic set of index t in a d-dimensional projective space P. If d is even then t d/2; if d is odd then t (d+1)/2.