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  "body": "\n## Statement\n\nLet $X$ be a smooth projective [[algebraic curve|XJM2MT4S3Y]] of genus $g$ over an algebraically closed field, with [[canonical divisor|W05VYY4HBR]] $K$.\nFor a [[divisor|Z9DN23DAE2]] $D$, let $\\ell(D)$ be the dimension of the space of rational functions $f$ with $\\operatorname{div}(f) + D \\ge 0$.\nThen the [[complete linear system|GHRPE8CXJ9]] $\\abs{D}$ has dimension $\\ell(D) - 1$.\nLet $D$ be an effective divisor that is special, meaning $\\ell(K - D) \u003e 0$.\nThen\n$$\\ell(D) - 1 \\le \\frac{\\deg D}{2}.$$\nEquality holds if and only if one of the following holds:\n\n1. $D = 0$;\n2. $D$ is linearly equivalent to $K$;\n3. $X$ is [[hyperelliptic|5SB8EKSQRN]], and $D$ is linearly equivalent to a multiple of a fibre of the degree-$2$ map $X \\to \\mathbb{P}^1$.\n\n## Remarks\n\nThe inequality comes from adding linear systems.\nSince $D$ and $K - D$ are both effective up to linear equivalence, the addition map $\\abs{D} \\times \\abs{K - D} \\to \\abs{K}$ is finite-to-one, so $\\dim \\abs{D} + \\dim \\abs{K - D} \\le \\dim \\abs{K} = g - 1$.\nAdding the [[Riemann–Roch theorem|WMVAT9MBZ1]] $\\ell(D) - \\ell(K - D) = \\deg D - g + 1$ gives $2(\\ell(D) - 1) \\le \\deg D$.\nThe hypothesis that $D$ is special is needed: for $\\deg D \u003e 2g - 2$, Riemann–Roch gives $\\ell(D) - 1 = \\deg D - g$, which exceeds $\\deg D / 2$ once $\\deg D \u003e 2g$.\nExample: on a hyperelliptic curve with degree-$2$ map divisor $g^1_2$, the divisor $D = r \\cdot g^1_2$ with $r \\le g - 1$ has $\\ell(D) = r + 1$ and $\\deg D = 2r$, so equality holds.\n\n## Explanations\n\n### L1 human\n\nA special divisor is one that is still small compared to the canonical divisor.\nClifford's theorem bounds how many functions can have poles bounded by such a divisor: the dimension of the linear system is at most half the degree.\nSo small divisors cannot move in large families.\nThe bound is reached only in trivial cases and on hyperelliptic curves.\n\n### L2 human\n\nRiemann–Roch computes $\\ell(D)$ exactly for nonspecial divisors, and Clifford's theorem is the first general bound in the special range $0 \\le \\deg D \\le 2g - 2$.\nIt is used to show that a non-hyperelliptic curve has a canonical embedding, and that a divisor of degree $2$ moving in a pencil forces the curve to be hyperelliptic.\nBrill–Noether theory refines the bound for a general curve, where the true dimensions are much smaller.\n",
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