Exercise 12, § 9

Let $\mathbf{P}(V)$ be a projective space (left) of finite dimension $n$ over a field $K$. Prove that there exists on the set of projective hyperplanes of $\mathbf{P}(V)$ a structure of projective space (right) of dimension $n$ over $K$, canonically isomorphic to that of $\mathbf{P}(V^*)$ ($V^*$ being the dual of $V$). If $M$ is a linear variety of dimension $r < n$ in $\mathbf{P}(V)$, deduce a structure of projective space of dimension $n - r - 1$ on the set of projective hyperplanes containing $M$. In particular, if $M$ is of dimension $n - 2$, one can define the cross ratio $\begin{bmatrix} H_1 & H_2 \\ H_4 & H_3 \end{bmatrix}$ of a quadruplet $(H_1, H_2, H_3, H_4)$ of distinct hyperplanes containing $M$. Prove that if $D \subset \mathbf{P}(V)$ is a line not meeting $M$, and if $a_i$ is the intersection of $D$ and of $H_i$ $(1 \leq i \leq 4)$, one has