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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==
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| {{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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Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.
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