Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 18 / 5 / i / Solution

Write the projective plane as $\mathbb{P}(V)$ for a three-dimensional complex vector space $V$. A projective line is the zero set of a nonzero linear functional $\ell \in V^{\ast }$. Two such functionals give the same line exactly when they differ by a nonzero scalar. Hence the parameter space of lines is the dual projective space $\mathbb{P}(V^{\ast })$. A choice of basis identifies $V^{\ast }$ with ${\mathbb{C}}^3$, giving $\boxed{({\mathbb{P}}^2{)}^{\vee }\cong {\mathbb{P}}^2}$. The description as $\mathbb{P}(V^{\ast })$ is intrinsic; the identification with the original $\mathbb{P}(V)$ requires a choice.