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Difference between revisions of "Clifford theorem"

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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224901.png" /> be a smooth projective curve over an algebraically closed field, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224902.png" /> be a divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224903.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224904.png" /> be the degree and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224905.png" /> the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224906.png" />. A positive divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224907.png" /> is called special if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224908.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c0224909.png" /> is the canonical divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249010.png" />. Clifford's theorem states: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249011.png" /> for any special divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249012.png" />, with equality if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249014.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249015.png" /> is a hyper-elliptic curve and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249016.png" /> is a multiple of the unique special divisor of degree 2 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249017.png" />. An equivalent statement of Clifford's theorem is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249019.png" /> is the linear system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249020.png" />. It follows from Clifford's theorem that the above inequality holds for any divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249022.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249024.png" /> is the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022490/c02249025.png" />.
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Let $X$ be a smooth projective curve over an algebraically closed field, and let $D$ be a divisor on $X$. 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 $X$.
  
 
====References====
 
====References====

Revision as of 15:37, 6 August 2014

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 $X$. 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 $X$.

References

[1] R.J. Walker, "Algebraic curves" , Springer (1978) MR0513824 Zbl 0399.14016
[2] N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)
[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


Comments

References

[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
How to Cite This Entry:
Clifford theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_theorem&oldid=32747
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article