Line bundle determined by a linear surjection $V \to V/W$

Line bundle determined by a linear surjection $V \to V/W$

I am new to things like line bundles and have just been reading page 10 of
this document here. Now I have some elementary questions:
Given a finite dimensional complex vector space $V$ and a one -
dimensional vector space $W$, we can form $V/W$ together with $\pi : V \to
V/W$ the canonical projection. Now I understand that upon taking
projectivizations we have a map $\Bbb{P}(\pi) : \Bbb{P}(V) - (\Bbb{P}(W) =
\{\ast\}) \longrightarrow \Bbb{P}(V/W)$. But why is this latter thing a
line bundle? I can see that given a line $L \subseteq V/W$, the inverse
image $\pi^{-1}(L)$ consists of all two dimensional subspaces $K$ in $V$
that contain $W$. But now how can I see that the fibre
$\Bbb{P}(\pi)^{-1}([L])$ "is" a one-dimensional vector space? Also, how
does one verify the local triviality condition?
Given a section $s$ of the map $\pi : V \to V/W$, we have a section
$\Bbb{P}(s)$ of the map $\Bbb{P}(\pi)$. However it is claimed in the
document I linked to that the image of $\Bbb{P}(V/W)$ in
$\Bbb{P}(V)\setminus \{\ast\}$ is a hypersurface. Why is this so?