Tautological line bundle

Remarks

For $V = k^2$, a point of $\mathbb{P}^1$ is $\ell = [a : b]$, and the fibre of $L$ over it is $\set{t(a, b) : t \in k}$. Over the open set $a \ne 0$, the map $(\ell, t(1, b/a)) \mapsto (\ell, t)$ trivializes $L$, so $L$ is locally a product with a line. The projection $L \to \mathbb{P}(V)$ is a line bundle but not the trivial one when $n \ge 1$. Indeed, $\mathcal{O}(-1)$ has no nonzero global sections, while a trivial bundle has a nowhere-zero section. The dual of $L$ corresponds to Serre's twisting sheaf $\mathcal{O}(1)$, whose global sections are the linear forms on $V$. Some authors write $\mathbb{P}(V)$ for the space of hyperplanes of $V$, which changes the sign conventions.