Clifford's theorem on special divisors

A divisor on a Riemann surface C is a formal sum $\backslash textstyle\; D\; =\; \backslash sum\_P\; m\_P\; P$ of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining $L(D)$ as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative coefficient, with at least that multiplicity. The dimension of $L(D)$ is finite, and denoted $\backslash ell(D)$. The linear system of divisors attached to D is the corresponding projective space of dimension $\backslash ell(D)-1$.

The other significant invariant of D is its degree d, which is the sum of all its coefficients.

A divisor is called special if ℓ(K − D) > 0, where K is the canonical divisor.Hartshorne p.296

Clifford's theorem states that for an effective special divisor D, one has:

:$2(\backslash ell(D)-\; 1)\; \backslash le\; d$,

and that equality holds only if D is zero or a canonical divisor, or if C is a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.

The Clifford index of C is then defined as the minimum of $d\; -\; 2(\backslash ell(D)\; -\; 1)$ taken over all special divisors (except canonical and trivial), and Clifford's theorem states this is non-negative. It can be shown that the Clifford index for a generic curve of genus g is equal to the floor function $\backslash lfloor\backslash tfrac\{g-1\}\{2\}\backslash rfloor.$

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.Eisenbud (2005) p.178

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, one defines the invariant a(C) in terms of 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_{i, i + 2} is zero. Green and Robert Lazarsfeld showed that a(C) + 1 is a lower bound for the Clifford index, and Green's conjecture states that equality always holds. There are numerous partial results.Eisenbud (2005) pp. 183-4.

Claire Voisin was awarded the Ruth Lyttle Satter Prize in Mathematics for her solution of the generic case of Green's conjecture in two papers.[http://www.math.polytechnique.fr/~voisin/Articlesweb/syzod.pdf Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin][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] The case of Green's conjecture for generic curves had attracted a huge amount of effort by algebraic geometers over twenty years before finally being laid to rest by Voisin.[http://www.agnesscott.edu/lriddle/women/prizes.htm#satter Satter Prize] The conjecture for arbitrary curves remains open.

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• {{cite book |first1=Enrico|last1= Arbarello |author1-link=Enrico Arbarello |first2=Maurizio |last2=Cornalba |first3=Phillip A.|last3= Griffiths |author3-link=Phillip Griffiths |first4=Joe|last4= Harris |author4-link=Joe Harris (mathematician)| title=Geometry of Algebraic Curves Volume I|series=Grundlehren de mathematischen Wisenschaften 267 | year =1985 | isbn=0-387-90997-4}}
• {{Citation | last1=Clifford | first1=William K. | author-link=William Kingdon Clifford | 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 | doi=10.1098/rstl.1878.0020}}
• {{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 }}
• {{cite book | first=William |last=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 }}
• {{cite book | first1=Phillip A.|last1= Griffiths | authorlink=Phillip Griffiths |first2=Joe|last2= Harris |authorlink2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=251 }}
• {{cite book | first=Robin|last= Hartshorne | authorlink=Robin Hartshorne | title=Algebraic Geometry | series=Graduate Texts in Mathematics | volume=52 | year=1977 | isbn=0-387-90244-9 }}

• {{eom|id=C/c022490|title=Clifford theorem|first=V.A.|last= Iskovskikh}}

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