Projective Geometry - Lecture Notes Chapter 2

Theorem 2.3.2: Let V be a vector space of dimension d+1 over the division ring F, and let P(V) be the corresponding projective space. Let H be a hyperplane of P(V). Then the homogeneous coordinates of the points of H are the solutions of a homogeneous equation with coefficients in F. Conversely, any homogeneous equation that is different from the "zero equation" describes a hyperplane of P(V).

Corollary 2.3.3: Any t-dimensional subspace U of a projective space of dimension d given by homogeneous coordinates can be described by a homogeneous system of d-t linear equations. More precisely, there exists a (d-t)×(d+1) matrix H such that a point P = (a₀:a₁: ...:a_d) is a point of U if and only if

Theorem 2.3.4: Let P = P(V) be a projective space whose points are given by homogeneous coordinates (a₀:a₁: ...:a_d) with a_i in F, (a₀,a₁, ...,a_d) (0,0, ..., 0). Then any hyperplane of P(V) can be represented by [b₀:b₁: ...:b_d] with b_j in F, [b₀,b₁, ...,b_d] [0,0, ...,0]. Conversely, to any such (d+1)-tuple [b₀:b₁: ...:b_d] there belongs a hyperplane. Moreover, we have

Corollary 2.3.5: Let P = P(V) be a coordinatized projective space. Then P P. In particular, P is coordinatized as well. Therefore the principle of duality holds for the class of all coordinatized projective spaces of fixed dimension d.

Theorem 2.3.6: Let be the hyperplane of P(V) with equation x₀ = 0. Then the affine space A = P\ can be described as follows:

• The points of A are the vectors (a₁,...,a_d) of the d-dimensional vector space F^d;
• the lines of A are the cosets of the 1-dimensional subspaces of F^d; that is the set u + <v>, where u,v in F^d, v 0;
• incidence is set-theoretical containment.