# Clifford theorem

2010 Mathematics Subject Classification: Primary: 14H51 [MSN][ZBL]

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 $X$ be a smooth projective curve over an algebraically closed field, and let $D$ be a divisor on $X$ (cf. Divisor (algebraic geometry)). Let $\deg D$ be the degree and $l(D)$ the dimension of $D$. A positive divisor $D$ is called special if $l(K-D)>0$, where $K$ is the canonical divisor on $X$. Clifford's theorem states: $\deg D\geq2l(D)-2$ for any special divisor $D$, with equality if $D=0$ or $D=K$ or if $X$ is a hyper-elliptic curve and $D$ is a multiple of the unique special divisor of degree 2 on $X$. An equivalent statement of Clifford's theorem is: $\dim|D|\leq(\deg D)/2$, where $|D|$ is the linear system of $D$. It follows from Clifford's theorem that the above inequality holds for any divisor $D$ on $X$ for which $0\leq\deg D\leq2g-2$, where $g=l(K)$ is the genus of $X$ (cf. Genus of a curve).

#### References

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