91Ó°ÊÓ

State and prove the dual of Pappus' theorem. [Hint: with care you can choose notation exactly dual to that in 5.10, e.g., \(p:(x=0), L=p \cap q=(0: 0: 1)\), etc.]

Prove that \(\mathbb{P}^{n}\) has a decomposition as a disjoint union of \(n+1\) subsets $$ \mathbb{P}^{n}=\\{\mathrm{pt}\\} \sqcup \mathbb{A}^{1} \sqcup \mathbb{A}^{2} \sqcup \cdots \sqcup \mathbb{A}^{n} $$ \(\left[\right.\) Hint: \(\mathbb{P}^{n}=\mathbb{A}^{n} \sqcup\) hyperplane at \(\left.\infty .\right]\)