Regarding the Defintion of Hirzebruch Surfaces $\mathbb{F}_n$ We have $\mathbb{F}_0= \mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ and, for $n \geqslant 1$, the $n-$th Hirzebruch Surfaces $\mathbb{F}_n$ is defined as a $\mathbb{C}\mathbb{P}^1$-bundle over $\mathbb{C}\mathbb{P}^1$ admitting a rational curve of self-intersection $-n$ as a section.
How can the statement (above) be understood?
 A: $\newcommand{\Hirzebruch}[1]{\mathbf{F}_{#1}}\newcommand{\Proj}{\mathbf{P}}$One approach (in fact, a natural definition of the Hirzebruch surfaces) is to start with the total space $L_{n}$ of the [sic, up to isomorphism] holomorphic line bundle of degree $n$ on $\Proj^{1}$, add the trivial line bundle, and projectivize:
$$
\Hirzebruch{n} = \Proj(L_{n} \oplus L_{0}).
$$
This construction adds a point at infinity in each fibre of $L_{n}$. The set of points at infinity is a copy of $\Proj^{1}$ with normal bundle $L_{-n}$, since
$$
\Hirzebruch{n} = \Proj(L_{n} \oplus L_{0})
\simeq \Proj(L_{n} \otimes(L_{0} \oplus L_{-n}))
\simeq \Proj(L_{0} \oplus L_{-n})
= \Hirzebruch{-n}.
$$
It's worth remarking that as smooth real $4$-manifolds the even Hirzebruch surfaces $\Hirzebruch{2k}$ are diffeomorphic to a product of spheres (analogous to a real $2$-torus), while the odd Hirzebruch surfaces $\Hirzebruch{2k+1}$ are diffeomorphic to the blow-up of the complex projective plane at a point (analogous to a Klein bottle). Kodaira's Complex Manifolds and Deformations of Complex Structures contains details.
