|
In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. == Statement == If ''D'' is a 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 functions, and in general lie in the function field of ''C''). Functions in this sense have a divisor of zeros and poles, counted with 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 :''L''(''D'') 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. 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〕 In this notation, Clifford's theorem is the statement that for an effective special divisor ''D'', :''ℓ''(''D'') − 1 ≤ ''d''/2, 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. The Clifford index of ''C'' is then defined as the minimum value of the ''d'' − 2''r''(''D''), taken over all special divisors (that are not canonical or trivial). Clifford's theorem is then the statement that this is non-negative. The Clifford index for a ''generic'' curve of genus ''g'' is the floor function of : 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〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Clifford's theorem on special divisors」の詳細全文を読む スポンサード リンク
|