Fundamental theorem of projective geometry

Remarks

A map $T$ with property 1 is called $\sigma$-semilinear. The proof fixes a frame of $n + 2$ points in general position and its image, and normalizes $T$ on the corresponding basis vectors. Comparing the points of a line through two frame points with the scalars defines $\sigma$. The configuration theorems that hold in $\mathbb{P}(V)$ show that $\sigma$ is additive and multiplicative. For $k = \R$ the identity is the only field automorphism, so $T$ is linear and $f$ is a projective transformation.

The hypothesis $n \ge 2$ is needed: $\mathbb{P}^1(k)$ has only one line, so every bijection of it is a collineation. The automorphism $\sigma$ cannot be dropped in general: complex conjugation of all coordinates is a collineation of $\mathbb{P}^2(\C)$ that is not a projective transformation. The theorem also holds for vector spaces over division rings. The collineation group of $\mathbb{P}^n(k)$, for $n \ge 2$, is therefore the group $\mathrm{P\Gamma L}(n + 1, k)$ of maps induced by semilinear bijections.

Going deeper

For a student

The theorem identifies the symmetry group of the incidence geometry $\mathbb{P}^n(k)$ with $\mathrm{P\Gamma L}(n + 1, k)$, which is $\mathrm{PGL}(n + 1, k)$ extended by the automorphisms of $k$. Together with the Veblen–Young theorem, it shows that the incidence structure of a projective space of dimension at least $3$ determines its coordinate division ring up to isomorphism. The fundamental theorem of affine geometry is the affine counterpart. It is the reason why, in real projective geometry, one may freely replace "collineation" by "projective transformation".