Difference between revisions of "Clifford theorem"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 4: | Line 4: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
Line 12: | Line 12: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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 |
Clifford theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_theorem&oldid=17287