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:
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
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.
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.
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.