View source for Clifford theorem

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" />.

====References====
<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====
<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>