---
tag: BDB06N91FN
kind: concept
name: Tautological line bundle
msc: [14C20, 14M15]
sources:
  - url: https://en.wikipedia.org/wiki/Tautological_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.

## Explanations

### L1 human

A point of projective space is a line through the origin of a vector space.
The tautological line bundle attaches to that point the line itself, so it is defined by a tautology.
On the real projective line, the result is a Möbius band, which shows that the bundle is twisted.
Over the complex numbers, the tautological bundle on $\mathbb{P}^n$ has no nonzero global sections for $n \ge 1$.