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  "body": "\n## Definition\n\nLet $k$ be a field, $V$ a [[vector space|05EGHEQZAW]] over $k$ of dimension $n + 1$, and $\\mathbb{P}(V)$ the [[projective space|ZT9A6JK8QF]] of lines in $V$.\nThe *tautological line bundle* on $\\mathbb{P}(V)$ is the closed subvariety\n$$L = \\set{(\\ell, v) \\in \\mathbb{P}(V) \\times V : v \\in \\ell}$$\ntogether with the projection $L \\to \\mathbb{P}(V)$.\nIts fibre over a point $\\ell$ is the line $\\ell \\subseteq V$ itself.\nIts sheaf of sections is a [[vector bundle|P4DZM975BV]] of rank $1$, isomorphic to $\\mathcal{O}_{\\mathbb{P}(V)}(-1)$.\n\n## Remarks\n\nFor $V = k^2$, a point of $\\mathbb{P}^1$ is $\\ell = [a : b]$, and the fibre of $L$ over it is $\\set{t(a, b) : t \\in k}$.\nOver 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.\nThe projection $L \\to \\mathbb{P}(V)$ is a line bundle but not the trivial one when $n \\ge 1$.\nIndeed, $\\mathcal{O}(-1)$ has no nonzero global sections, while a trivial bundle has a nowhere-zero section.\nThe dual of $L$ corresponds to [[Serre's twisting sheaf|V93Y0XYPZD]] $\\mathcal{O}(1)$, whose global sections are the linear forms on $V$.\nSome authors write $\\mathbb{P}(V)$ for the space of hyperplanes of $V$, which changes the sign conventions.\n",
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