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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 form of a ''linear system'' of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors ''D'' on a general scheme or even a ringed space (X, OX).〔Grothendieck, Alexandre; Dieudonné, Jean. ''EGA IV'', 21.3.〕 A linear system of dimension 1, 2, or 3 is called a pencil, a net, or a web. ==Definition == Given the fundamental idea of a rational function on a general variety ''V'', or in other words of a function ''f'' in the function field of ''V'', divisors D and E are ''linearly equivalent'' if : where (''f'') denotes the divisor of zeroes and poles of the function ''f''. Note that if ''V'' has singular points, 'divisor' is inherently ambiguous (Cartier divisors, Weil divisors: see divisor (algebraic geometry)). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below. A complete linear system on ''V'' is defined as the set of all effective divisors linearly equivalent to some given divisor ''D''. It is denoted ''|D|''. Let ''L(D)'' be the line bundle associated to ''D''. In the case that ''V'' is a nonsingular projective variety the set ''|D|'' is in natural bijection with 〔Hartshorne, R. 'Algebraic Geometry', proposition II.7.7, page 157〕 and is therefore a projective space. A linear system is then a projective subspace of a complete linear system, so it corresponds to a vector subspace ''W'' of The dimension of the linear system is its dimension as a projective space. Hence . Since a Cartier divisor class is an isomorphism class of a line bundle, linear systems can also be introduced by means of the line bundle or invertible sheaf language, without reference to divisors at all. In those terms, divisors ''D'' (Cartier divisors, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic. 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「linear system of divisors」の詳細全文を読む スポンサード リンク
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