Duality in projective geometry

Remarks

The axioms of a projective plane are self-dual, so $\Pi^{*}$ is again a projective plane. Hence the principle of duality: if a statement holds in every projective plane, so does its dual.

Example: $\mathbb{P}^2(k)$ over a field $k$ has a duality sending the point $[a : b : c]$ to the line $a x_0 + b x_1 + c x_2 = 0$ and the line $a x_0 + b x_1 + c x_2 = 0$ to the point $[a : b : c]$. Check: the point $[x_0 : x_1 : x_2]$ lies on the line with coefficients $(a, b, c)$ exactly when $a x_0 + b x_1 + c x_2 = 0$, a condition symmetric in the two triples, so incidence is preserved in both directions.

Non-example: the dual of an affine plane is not an affine plane. Two parallel lines become two points with no line through both, which fails the first axiom of an affine plane.

Let $\mathbb{P}(V)$ be a projective space of finite dimension $n$. The map $W \mapsto W^{0}$, sending a subspace to its annihilator in the dual space, reverses inclusion and exchanges projective dimensions $d$ and $n - 1 - d$.