Linear system of divisors: Difference between revisions

Revision as of 04:09, 22 July 2020

Revision as of 12:11, 19 November 2020

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Note that if <math>X</math> has [[Mathematical singularity|singular points]], 'divisor' is inherently ambiguous ([[Cartier divisor]]s, [[Weil divisor]]s: see [[divisor (algebraic geometry)]]). The definition in that case is usually said with greater care (using [[invertible sheaves]] or [[holomorphic line bundle]]s); see below.

A '''complete linear system''' on <math>X</math> is defined as the set of all effective divisors linearly equivalent to some given divisor <math>D \in \text{Div}(X)</math>. It is denoted <math>|D|</math>. Let <math>\mathcal{L}(D)</math> be the line bundle associated to <math>D</math>. In the case that <math>X</math> is a nonsingular projective variety the set <math>|D|</math> is in natural bijection with <math> (\Gamma(VX,\mathcal{L}) \smallsetminus \{0\})/k^\ast, </math> <ref name=":0">Hartshorne, R. 'Algebraic Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342</ref> {{explain|date=July 2020}} and is therefore a projective space.

A '''linear system''' <math> \mathfrak{d} </math> is then a projective subspace of a complete linear system, so it corresponds to a vector subspace ''W'' of <math> \Gamma(VX,\mathcal{L}). </math> The dimension of the linear system <math> \mathfrak{d} </math> is its dimension as a projective space. Hence <math> \dim \mathfrak{d} = \dim W - 1 </math>.

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 <math>D</math> ([[Cartier divisor]]s, to be precise) correspond to line bundles, and '''linear equivalence''' of two divisors means that the corresponding line bundles are isomorphic.