Projective Geometry I - Lecture Notes

8.1 Pn(R)

Def: Pn(R), for R a division ring.

Points of vector space Rn+1 except for the zero vector, with the equivalence, (x1, ..., xn+1) ~ (x1, ..., xn+1)µ = (x1µ, ..., xn+1µ). (N.B. Multiply on the right by µ ).

Examples (with n = 2): With R = Z2 we get PG(2,2), the Fano Plane.

With R = GF(q), we get PG(2,q) having q2 + q + 1 points.

With R = R , we get the real projective plane.

Theorem 8.5 : Pn(R) over a division ring always satisfies Desargues' axiom P5.

8.2 The Automorphism Group of Pn(R)

Def: semi-linear transformation T, satisfies
T(x + y) = T(x) + T(y) , for all x,y in Pn(R) and
for which there exists a ring automorphism þ : R --> R such that
T(µ x) = þ(µ )T(x) for all µ in R.

Def: an invertible square matrix over a division ring.

For fields, these are just the matrices with non-zero determinant. Over general division rings, determinants do not make sense. (** Hockey puck, they make sense, they just don't have the nice properties that you want).

Prop 8.8: If A is an invertible n+1 x n+1 matrix over R, then Ax = x' defines an automorphism of Pn(R).

Lemma 8.9: TA and TA' have the same effect on the vertices of the standard (n+1)-simplex iff there exists a µ in R, µ not 0, such that A' = Aµ.

Lemma 8.10: TµI where µI is a diagonal matrix, is the identity transformation of Pn(R) iff µ is in the center of R. Otherwise, it is the automorphism given by (x1, ..., xn+1) --> (þ(x1), ...,þ(xn+1)), where þ(x) = µxµ-1, i.e. an inner automorphism.

Def: PGL(n, R) - the projective general linear group over R.

Prop. 8.12: TA = TA' iff there exists a nonzero µ in the center of R such that A' = Aµ .

Thm 8.13: Let A0, ..., An+1 and B00, ..., Bn+1 be two n+2 -tuples of points, no n+1 of which lie in a hyperplane. Then there is an automorphism T in PGL(n, R) such that T(Ai) = Bi, for all i. If R is a field, then T is unique.

Lemma 8.14: n+1 points in Pn(R) are noncollinear iff the matrix formed by their coordinates as columns is invertible.

Prop 8.15: Let {\phi} be any automorphism of Pn(R) which leaves fixed the standard simplex of points. Then there is an automorphism of R, þ, such that {\phi}(x1, ..., xn+1) = (þ(x1), ..., þ(xn+1)).

Def: Let {\phi} above be renamed Sþ.

Prop. 8.16: The mapping {\Psi} : Aut R --> Aut Pn(R) given by þ --> Sþ is an isomorphism of Aut R onto the subgroup H of Aut Pn(R) consisting of those automorphisms which fix the standard simplex.

Thm 8.18: PGL(n, R) and H generate Aut Pn(R). The intersection of PGL(n,R) and H is isomorphic to the group of inner automorphisms of R.

Prop 8.19: The identity is the only automorphism of R .

Thm 8.20: PGL(n, R ) = Aut Pn(R ).

Thm 8.22: Given two ordered sets, X1 and X2, of n+2 points of Pn(R ) in general position (no n + 1 points lie in a hyperplane), there is a unique automorphism of Pn(R ) sending X1 to X2.

8.3 The Algebraic Meaning of Axioms P6 and P7

Prop 8.23: Fano's axiom P6 holds in P2(R) iff the characteristic of R is not 2.

Thm 8.24 (Hilbert): The fundamental axiom P7 holds in P2(R) iff R is commutative.

8.4 Independence of Axioms

P7 implies P5, but in the presence of P1-P4, the other axioms are independent.