mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Clifford's theorem on special divisors is a result of on

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s, showing the constraints on special linear systems on a curve ''C''.

Statement

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

on a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

''C'' is a

formal sum In mathematics, a formal sum, formal series, or formal linear combination may be: *In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. *In linear algebra, an ...

\textstyle D = \sum_P m_P P

of points ''P'' on ''C'' with integer coefficients. One considers a divisor as a set of constraints on

meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...

s 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

\ell(D)

. The

linear system of divisors In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the f ...

attached to ''D'' is the corresponding

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

of dimension

\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 The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical examp ...

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

2(\ell(D)- 1) \le d

, and that equality holds only if ''D'' is zero or a canonical divisor, or if ''C'' is a

hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...

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(\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 Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

''g'' is equal to the

floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

\lfloor\tfrac\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 mathematics, the gonality of an algebraic curve ''C'' is defined as the lowest degree of a nonconstant rational map from ''C'' to the projective line. In more algebraic terms, if ''C'' is defined over the field ''K'' and ''K''(''C'') denotes th ...

: in many cases the Clifford index is equal to the gonality minus 2.Eisenbud (2005) p.178

Green's conjecture

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 In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is ...

has linear syzygies. In detail, one defines the invariant ''a''(''C'') in terms of the minimal

free resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to de ...

of the

homogeneous coordinate ring In algebraic geometry, the homogeneous coordinate ring is a certain commutative ring assigned to any projective variety. If ''V'' is an algebraic variety given as a subvariety of projective space of a given dimension ''N'', its homogeneous coordina ...

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. 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.Satter Prize
/ref> The conjecture for ''arbitrary'' curves remains open.

Notes

References

* * * * * *

External links

* {{Algebraic curves navbox Algebraic curves Theorems in algebraic geometry Unsolved problems in geometry