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  "body": "\n## Summary\n\nLet $\\mathbb{P}(V)$ be the projective space of lines in a vector space $V$.\nThe tautological line bundle on $\\mathbb{P}(V)$ is the line bundle whose fibre over a point is the line that the point represents.\nIts sheaf of sections is $\\mathcal{O}(-1)$, the dual of Serre's twisting sheaf.\nIt 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.\n\n## Explanations\n\n### L1 human\n\nA point of projective space is a line through the origin of a vector space.\nThe tautological line bundle attaches to that point the line itself, so it is defined by a tautology.\nOn the real projective line, the result is a Möbius band, which shows that the bundle is twisted.\nOver the complex numbers, the tautological bundle on $\\mathbb{P}^n$ has no nonzero global sections for $n \\ge 1$.\n",
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