Let \(D\) be a divisor on a smooth projective curve \(C\) of genus \(g\). The index of speciality of \(D\) is \(\ell(K-D) = h^1(\OO_C(D))\). \(D\) is special if this is positive, and nonspecial if it vanishes.
If \(\deg D > 2g-2\) then \(D\) is nonspecial, and then Riemann–Roch reads \[ \ell(D) = \deg D + 1 - g . \]
Riemann–Roch expresses \(\ell(D)\) as \(\deg D+1-g+\ell(K-D)\). Once \(\deg D\) exceeds \(\deg K = 2g-2\), the divisor \(K-D\) has negative degree, so \(\ell(K-D)=0\). For every \(D\), Riemann–Roch gives the inequality \(\ell(D) \geq \deg D + 1 - g\).
The canonical divisor \(K\) is special, since \(\ell(K-K)=1\), and on a hyperelliptic curve of genus \(g\ge2\) every divisor of the \(g^1_2\) is special. For a special effective divisor \(D\), Clifford’s theorem gives \(\ell(D)-1\le\frac12\deg D\), with equality if and only if \(D=0\), \(D\sim K\), or \(C\) is hyperelliptic and \(D\) is a multiple of the \(g^1_2\) [@Har10a, Theorem IV.5.4].
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T-MWDVLRiemann--Roch and Serre duality on a curveuses
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T-CRVCLIFFClifford's theoremusesPR-CRVDEGBDThe degree thresholds $2g$ and $2g+1$uses