A theorem establishing an inequality between the degree and the dimension of a special divisor on an algebraic curve. It was proved by W. Clifford.
Let be a smooth projective curve over an algebraically closed field, and let be a divisor on . Let be the degree and the dimension of . A positive divisor is called special if , where is the canonical divisor on . Clifford's theorem states: for any special divisor , with equality if or or if is a hyper-elliptic curve and is a multiple of the unique special divisor of degree 2 on . An equivalent statement of Clifford's theorem is: , where is the linear system of . It follows from Clifford's theorem that the above inequality holds for any divisor on for which , where is the genus of .
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Clifford theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Clifford_theorem&oldid=23784