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  "body": "\n## Statement\n\nLet $k$ be a field and let $X$ be a variety that is [[smooth|EXKGG9X6J0]] over $k$.\nLet $Y \\subseteq X$ be a closed subvariety that is smooth over $k$ and of codimension $1$.\nThen $Y$ is an effective [[Cartier divisor|FHV05ZK99F]] on $X$, with [[invertible sheaf|7JCF2TE3V5]] $\\mathcal{O}_X(Y)$.\nThe [[canonical bundles|W05VYY4HBR]] satisfy\n$$\\omega_Y \\cong \\left(\\omega_X \\otimes \\mathcal{O}_X(Y)\\right)|_Y.$$\nIn terms of divisors, $K_Y = (K_X + Y)|_Y$.\n\n## Remarks\n\nThe idea of the proof: for $Y$ smooth with ideal sheaf $\\mathcal{I}$, the conormal sequence $0 \\to \\mathcal{I}/\\mathcal{I}^2 \\to \\Omega_X|_Y \\to \\Omega_Y \\to 0$ is an exact sequence of vector bundles.\nTaking top exterior powers gives $\\omega_X|_Y \\cong \\omega_Y \\otimes \\mathcal{I}/\\mathcal{I}^2$.\nSince $\\mathcal{I} = \\mathcal{O}_X(-Y)$, the conormal bundle is $\\mathcal{I}/\\mathcal{I}^2 \\cong \\mathcal{O}_X(-Y)|_Y$, and the formula follows.\n\nExample: let $C \\subseteq \\mathbb{P}^2_k$ be a smooth plane curve of degree $d$.\nThen $K_{\\mathbb{P}^2} = -3H$ for a line $H$ and $C \\sim dH$, so $K_C = (d-3)H|_C$ has degree $d(d-3)$.\nSince $\\deg K_C = 2g - 2$, this gives the genus $g = (d-1)(d-2)/2$.\n\nFor a smooth closed subvariety $Y$ of any codimension $r$, the same argument gives $\\omega_Y \\cong \\omega_X|_Y \\otimes \\det N_{Y/X}$, where $N_{Y/X}$ is the normal bundle.\n\n## Explanations\n\n### L1 human\n\nThe adjunction formula computes the canonical bundle of a hypersurface from that of the ambient space.\nOne only needs to add the hypersurface itself, viewed as a divisor, and restrict.\nThis turns questions about a subvariety into questions about the bigger space, where computations are often easy.\n\n### L2 human\n\nThe formula is the standard way to compute the genus of curves on surfaces and the canonical class of complete intersections.\nFor a smooth hypersurface of degree $d$ in $\\mathbb{P}^n$ it gives $\\omega_Y \\cong \\mathcal{O}_Y(d - n - 1)$, which decides whether $Y$ is Fano, Calabi–Yau or of general type.\nOn a surface $X$ it gives the genus formula $2g - 2 = C \\cdot (C + K_X)$ for a smooth curve $C$.\n",
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