Exercise 12, § 9

Let $\mathbf{P}(V)$ be a (left) projective space of finite dimension $n$ over a field $K$. Show that there exists on the set of projective hyperplanes of $\mathbf{P}(V)$ an $n$-dimensional (right) projective space structure 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 $\mathbf{P}(V)$, derive an $(n - r - 1)$-dimensional projective space structure on the set of projective hyperplanes containing $M$. In particular if $M$ is of

dimension $n - 2$, the cross-ratio $\begin{bmatrix} H_1 & H_2 \\ H_4 & H_3 \end{bmatrix}$ of a quadruple $(H_1, H_2, H_3, H_4)$ of distinct hyperplanes containing M can be defined. Show that if $D \subset \mathbf{P}(V)$ is a line not meeting M and $a_i$ is the intersection of D with $H_i$ ($1 \leq i \leq 4$), then