Content deleted Content added
m Dating maintenance tags: {{Expand section}} |
TakuyaMurata (talk | contribs) |
||
Line 63:
== A map determined by a linear system ==
<!-- the below is a coordinate-free approach; while useful and important in application, we should also give a less abstract construction as well. -->
Each linear system on an algebraic variety determines a morphism from the complement of the base locus to a projective space of dimension of the system, as follows. (In a sense, the converse is also true;
Let ''L'' be a line bundle on an algebraic variety ''X'' and <math>V \subset \Gamma(X, L)</math> a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when ''V'' is base-point-free; in other words, the natural map <math>V \otimes_k \mathcal{O}_X \to L</math> is surjective (here, ''k'' = the base field). Or equivalently, <math>\operatorname{Sym}((V \otimes_k \mathcal{O}_X) \otimes_{\mathcal{O}_X} L^{-1}) \to \bigoplus_{n=0}^{\infty} \mathcal{O}_X</math> is surjective. Hence, writing <math>V_X = V \times X</math> for the trivial vector bundle and passing the surjection to the [[relative Proj]], there is a [[closed immersion]]:
| |||