Exercise 16, § 9

(a) Let $E = \mathbf{P}(V)$ and $E' = \mathbf{P}(V')$ be two projective spaces of the same dimension $n \geq 2$ over two fields K, $K'$ respectively and let $u$ be a bijective mapping of E onto $E'$, mapping any three points on a straight line to three points on a straight line. Show that there exist an isomorphism $\sigma$ of K onto $K'$ and a bijective semi-linear mapping $v$ of V onto $V'$ (relative to $\sigma$) such that $u$ is the mapping obtained from $v$ when passing to the quotients ("fundamental theorem of projective geometry"; use Exercise 7). Suppose also that $V' = V$ and K is commutative; for $u$ to be a projective mapping, it is necessary and sufficient that also

for every quadruple $(a, b, c, d)$ of distinct points on a straight line in $\mathbf{P}(V)$.

(b) Let $p$ be an integer such that $1 \leq p \leq n - 1$. Show that the first conclusion of (a) holds when the image under $u$ of every $p$-dimensional projective linear variety is contained in a $p$-dimensional projective linear variety.