Tautological line bundle

Summary

Let $\mathbb{P}(V)$ be the projective space of lines in a vector space $V$. The tautological line bundle on $\mathbb{P}(V)$ is the line bundle whose fibre over a point is the line that the point represents. Its sheaf of sections is $\mathcal{O}(-1)$, the dual of Serre's twisting sheaf. It is universal: a line bundle generated by finitely many global sections is pulled back from its dual $\mathcal{O}(1)$ by a map to projective space.