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| In [[mathematics]], '''Clifford's theorem on special divisors''' is a result of {{harvs|txt|authorlink=William Kingdon Clifford|first=W. K. |last=Clifford|year=1878}} on [[algebraic curve]]s, showing the constraints on [[special linear system]]s on a curve ''C''.
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| == Statement ==
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| If ''D'' is a [[divisor on an algebraic curve|divisor]] on ''C'', then ''D'' is (abstractly) a [[formal sum]] of points ''P'' on ''C'' (with integer coefficients), and in this application a set of constraints to be applied to functions on ''C'' (if ''C'' is a [[Riemann surface]], these are [[meromorphic function]]s, and in general lie in the [[function field of an algebraic variety|function field]] of ''C''). Functions in this sense have a divisor of zeros and poles, counted with [[Multiplicity (mathematics)|multiplicity]]; a divisor ''D'' is here of interest as a set of constraints on functions, insisting that poles at given points are ''only as bad'' as the positive coefficients in ''D'' indicate, and that zeros at points in ''D'' with a negative coefficient have ''at least'' that multiplicity. The dimension of the vector space
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| :''L''(''D'')
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| of such functions is finite, and denoted ''ℓ''(''D''). Conventionally the [[linear system of divisors]] attached to ''D'' is then attributed dimension ''r''(''D'') = ''ℓ''(''D'') − 1, which is the dimension of the [[projective space]] parametrizing it.
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| The other significant invariant of ''D'' is its degree, ''d'', which is the sum of all its coefficients.
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| A divisor is called ''[[special divisor|special]]'' if ''ℓ''(''K'' − ''D'') > 0, where ''K'' is the [[canonical divisor]].<ref>Hartshorne p.296</ref>
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| In this notation, '''Clifford's theorem''' is the statement that for an effective [[special divisor]] ''D'',
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| :''ℓ''(''D'') − 1 ≤ ''d''/2,
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| together with the information that the case of equality here is only for ''D'' zero or canonical, or ''C'' a [[hyperelliptic curve]] and ''D'' linearly equivalent to an integral multiple of a hyperelliptic divisor.
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| ==Clifford index==
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| The '''Clifford index''' of ''C'' is then defined as the minimum value of the ''d'' − 2''r''(''D''), taken over all special divisors. Clifford's theorem is then the statement that this is non-negative. The Clifford index for a ''generic'' curve of [[genus (mathematics)|genus]] ''g'' is the [[floor function]] of
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| :<math>\frac{g-1}{2}.</math>
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| The Clifford index measures how far the curve is from being hyperelliptic. It may be thought of as a refinement of the [[gonality]]: in many cases the Clifford index is equal to the gonality minus 2.<ref name=Eis178>Eisenbud (2005) p.178</ref>
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| ==Green's Conjecture==
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| A conjecture of Mark Green states that the Clifford index for a curve over the complex numbers that is not hyperelliptic should be determined by the extent to which ''C'' as [[canonical curve]] has linear syzygies. In detail, the invariant ''a''(''C'') is determined by the minimal [[free resolution]] of the [[homogeneous coordinate ring]] of ''C'' in its canonical embedding, as the largest index ''i'' for which the [[graded Betti number]] β<sub>''i'', ''i'' + 2</sub> is zero. Green and Lazarsfeld showed that ''a''(''C'') + 1 is a lower bound for the Clifford index, and '''Green's conjecture''' is that equality always holds. There are numerous partial results.<ref name=Eis1834>Eisenbud (2005) pp. 183-4.</ref>
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| [[Claire Voisin]] was awarded the [[Ruth Lyttle Satter Prize in Mathematics]] for her solutions of two long standing mathematical problems, "the Green's conjecture (Green's canonical [[syzygy (mathematics)|syzygy]] conjecture for generic curves of odd genus),<ref>[http://www.math.polytechnique.fr/~voisin/Articlesweb/syzod.pdf Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin]</ref> and Green's generic syzygy conjecture for curves of even genus lying on a K3 surface<ref>[http://www.math.polytechnique.fr/~voisin/Articlesweb/syzy.pdf Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface - Claire Voisin]</ref>". Green's conjecture attracted a huge amount of effort by algebraic geometers over twenty years before finally being laid to rest by Voisin.<ref>[http://www.agnesscott.edu/lriddle/women/prizes.htm#satter Satter Prize]</ref>
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| ==Notes== | |
| {{Reflist}}
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| ==References==
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| *{{cite book | author=E. Arbarello | coauthors=M. Cornalba, P.A. Griffiths, J. Harris|title=Geometry of Algebraic Curves Volume I|series=Grundlehren de mathematischen Wisenschaften 267 | year =1985 | isbn=0-387-90997-4}}
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| *{{Citation | last1=Clifford | first1=W. K. | title=On the Classification of Loci | jstor=109316 | publisher=The Royal Society | year=1878 | journal=[[Philosophical Transactions of the Royal Society of London]] | issn=0080-4614 | volume=169 | pages= 663–681}}
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| *{{cite book |authorlink=David Eisenbud |first=David |last=Eisenbud |title=The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry | series=[[Graduate Texts in Mathematics]] | volume=229 | year=2005 |location=New York, NY |publisher=[[Springer-Verlag]] | isbn=0-387-22215-4 | zbl=1066.14001 }}
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| *{{cite book | author=William Fulton | authorlink=William Fulton (mathematician) | title=Algebraic Curves | series=Mathematics Lecture Note Series | publisher=W.A. Benjamin | year=1974 | isbn=0-8053-3080-1 | page=212 }}
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| * {{cite book | author=P.A. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=251 }}
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| * {{cite book | author=Robin Hartshorne | authorlink=Robin Hartshorne | title=Algebraic Geometry | series=[[Graduate Texts in Mathematics]] | volume=52 | year=1977 | isbn=0-387-90244-9 }}
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| ==External links==
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| *{{eom|id=C/c022490|title=Clifford theorem|first=V.A.|last= Iskovskikh}}
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| {{Algebraic curves navbox}}
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| [[Category:Algebraic curves]]
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| [[Category:Theorems in algebraic geometry]]
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