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  "body": "\n## Definition\n\nLet $X$ be a Noetherian [[scheme|5A35TYZD02]] and let $Z \\subseteq X$ be a closed subscheme with [[ideal sheaf|N3PGYJJQ7W]] $\\mathcal{I} \\subseteq \\mathcal{O}_X$.\nThe *blow-up* of $X$ along $Z$ is the scheme\n$$\\operatorname{Bl}_Z X = \\operatorname{Proj}_X \\bigoplus_{d \\ge 0} \\mathcal{I}^d,$$\ngiven by the relative [[Proj construction|A39XQHCVNW]], with $\\mathcal{I}^0 = \\mathcal{O}_X$, together with its projection $\\pi \\colon \\operatorname{Bl}_Z X \\to X$.\nThe closed subscheme $E = \\pi^{-1}(Z)$ is the *exceptional divisor*, and $Z$ is the *centre* of the blow-up.\n\n## Remarks\n\nThe ideal sheaf $\\pi^{-1}\\mathcal{I} \\cdot \\mathcal{O}_{\\operatorname{Bl}_Z X}$ of $E$ is invertible, so $E$ is an effective [[Cartier divisor|FHV05ZK99F]].\nThe morphism $\\pi$ is projective, and it is an isomorphism over $X \\setminus Z$.\nIt has a universal property: every morphism $f \\colon Y \\to X$ such that $f^{-1}\\mathcal{I} \\cdot \\mathcal{O}_Y$ is invertible factors uniquely through $\\pi$.\n\nExample: the blow-up of $\\mathbb{A}^2 = \\operatorname{Spec} k[x, y]$ at the origin, with $\\mathcal{I} = (x, y)$, is the closed subvariety\n$$\\set{((x, y), [u : v]) \\in \\mathbb{A}^2 \\times \\mathbb{P}^1 : x v = y u}.$$\nOver a point $(x, y) \\ne (0, 0)$ the condition forces $[u : v] = [x : y]$, so $\\pi$ is bijective there.\nOver the origin every $[u : v]$ is allowed, so $E \\cong \\mathbb{P}^1$, the [[projective line|ZT9A6JK8QF]] of directions through the origin.\n\nNon-example: the projection $\\mathbb{A}^2 \\times \\mathbb{P}^1 \\to \\mathbb{A}^2$ is not the blow-up at the origin.\nIt fails to be an isomorphism over $\\mathbb{A}^2 \\setminus \\set{0}$, since every fibre is a whole $\\mathbb{P}^1$.\n\nBlowing up an effective Cartier divisor changes nothing: $\\pi$ is then an isomorphism.\n",
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