Clifford theorem

2020 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

[1]	R.J. Walker, "Algebraic curves" , Springer (1978) MR0513824 Zbl 0399.14016
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[3]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[4]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2]	E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985) MR0770932 Zbl 0559.14017