Linear system of divisors: Difference between revisions

Revision as of 15:46, 24 July 2019

Revision as of 05:04, 2 August 2019

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Finally, when a basis of ''V'' is chosen, the above discussion becomes more down-to-earth (and that is the style used in Hartshorne, Algebraic Geometry).<!-- and we should give that version here as well. -->

== Linear system determined by a map to a projective space ==

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Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably.

In general, (under some assumptions<!-- need a ref -->), one can pullback a linear system as follows: let <math>f: X \to Y</math> be a morphism of algebraic varieties. Then the pullback of a linear system <math>\mathfrak{d}</math> on ''Y'' is <math>f^{-1}(\mathfrak{d}) = \{ f^{-1}(D) | D \in \mathfrak{d} \}.</math>

==References==