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Difference between revisions of "Clifford theorem"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.J. Walker,   "Algebraic curves" , Springer (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.G. Chebotarev,   "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.J. Walker, "Algebraic curves" , Springer (1978) {{MR|0513824}} {{ZBL|0399.14016}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
  
  
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello,   M. Cornalba,   P.A. Griffiths,   J.E. Harris,   "Geometry of algebraic curves" , '''1''' , Springer (1985)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , '''1''' , Springer (1985) {{MR|0770932}} {{ZBL|0559.14017}} </TD></TR></table>

Revision as of 21:50, 30 March 2012

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 .

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=17287
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article