Fundamental theorem of projective geometry

Statement

Let $V$ and $W$ be vector spaces of dimension at least $3$ over fields $k$ and $\ell$ . If a bijection

f:\mathbb P(V)\longrightarrow\mathbb P(W)

maps every projective line onto a projective line, then there are a field isomorphism $\sigma:k\to\ell$ and a bijective $\sigma$ -semilinear map $T:V\to W$ such that

f([v])=[T(v)].

The map $T$ is unique up to multiplication by a nonzero scalar in $\ell$ , and $\sigma$ is unique.

Collineation form

A bijection that maps projective lines onto projective lines is called a collineation. For a fixed $k$ -vector space $V$ of dimension at least $3$ , the theorem says that every collineation belongs to the $\operatorname{P\Gamma L}(V)$ . Those induced by genuinely linear maps form the subgroup $\operatorname{PGL}(V)$ .

The statement can equivalently require preservation of all projective subspaces or preservation of incidence in both directions. Once projective lines are preserved, higher-dimensional projective subspaces are preserved because they are generated by points and lines.

Why projective dimension one is excluded

If $\dim V=2$ , then $\mathbb P(V)$ is itself a projective line and has no proper projective lines on which to test incidence. Every permutation of its points is line-preserving in that vacuous sense, and most permutations do not arise from semilinear maps. Thus the hypothesis $\dim V\ge3$ , equivalently projective dimension at least $2$ , is essential.

References

Emil Artin, Geometric Algebra, Interscience, 1957. Relevant: Theorem 2.26, semilinear realization of projective collineations.
Peter J. Cameron, Projective and Polar Spaces, Queen Mary and Westfield College, 1992. Author-maintained text. Relevant: Chapter 1, the fundamental theorem.