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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $X$ be a smooth projective [[algebraic curve|XJM2MT4S3Y]] over $k$ of [[genus|6W8TDV9VVN]] $g$.\nLet $k(X)$ be the field of rational functions on $X$.\nFor a [[divisor|Z9DN23DAE2]] $D$ on $X$, let\n$$L(D) = \\set{f \\in k(X)^\\times : \\operatorname{div}(f) + D \\ge 0} \\cup \\set{0},$$\nwhich is a finite-dimensional vector space over $k$, and let $\\ell(D) = \\dim_k L(D)$.\nLet $K$ be a [[canonical divisor|W05VYY4HBR]] of $X$.\nThen for every divisor $D$ on $X$,\n$$\\ell(D) - \\ell(K - D) = \\deg D - g + 1.$$\n\n## Remarks\n\nIn sheaf language, $\\ell(D) = h^0(X, \\mathcal{O}_X(D))$ and $\\ell(K - D) = h^1(X, \\mathcal{O}_X(D))$ by Serre duality, so the formula reads $\\chi(\\mathcal{O}_X(D)) = \\deg D + 1 - g$.\nOne proof checks the formula for $D = 0$, where it says $h^1(\\mathcal{O}_X) = g$.\nThen it shows that both sides change by $1$ when a point $P$ is added to $D$.\nThis uses the exact sequence $0 \\to \\mathcal{O}_X(D) \\to \\mathcal{O}_X(D + P) \\to k_P \\to 0$.\nTaking $D = 0$ and $D = K$ gives $\\ell(K) = g$ and $\\deg K = 2g - 2$.\nIf $\\deg D \u003e 2g - 2$, then $\\deg(K - D) \u003c 0$, so $\\ell(K - D) = 0$ and $\\ell(D) = \\deg D - g + 1$ exactly.\nExample: on the projective line, $g = 0$ and $K = -2P$ for any point $P$.\nFor $D = nP$ with $n \\ge 0$ the formula gives $\\ell(D) = n + 1$.\nThis is the dimension of the space of polynomials of degree at most $n$ in an affine coordinate.\n\n## Explanations\n\n### L1 human\n\nA divisor prescribes where a function may have poles and how bad they may be, and where it must vanish.\nThe theorem counts how many independent functions obey such a prescription.\nThe answer is the degree of the prescription, corrected by the genus of the curve and by a second count that involves the canonical divisor.\nFor prescriptions of large degree the correction term disappears, and the count is simply $\\deg D - g + 1$.\n\n### L2 human\n\nRiemann–Roch is the main computational tool of the theory of curves.\nIt shows that a divisor of degree at least $2g + 1$ gives an embedding into projective space.\nIt also shows that every curve of genus $0$ is the projective line, and that every curve of genus $1$ with a point is a plane cubic.\nIt feeds directly into [[Clifford's theorem|RSW29AVKGM]], the [[Riemann–Hurwitz formula|42A5ZT6AWN]] and Brill–Noether theory.\nIts higher-dimensional generalizations are the Hirzebruch–Riemann–Roch and Grothendieck–Riemann–Roch theorems.\n",
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