Editing Line bundle (section)

===Maps to projective space===
Suppose that <math>X</math> is a space and that <math>L</math> is a line bundle on <math>X</math>. A '''global section''' of <math>L</math> is a function <math>s:X\to L</math> such that if <math>p:L\to X</math> is the natural projection, then <math>p\circ s=\operatorname{id}_X</math>.  In a small neighborhood <math>U</math> in <math>X</math> in which <math>L</math> is trivial, the total space of the line bundle is the product of <math>U</math> and the underlying field <math>k</math>, and the section <math>s</math> restricts to a function <math>U\to k</math>. However, the values of <math>s</math> depend on the choice of trivialization, and so they are determined only up to multiplication by a nowhere-vanishing function.

Global sections determine maps to projective spaces in the following way: Choosing <math>r+1</math> not all zero points in a fiber of <math>L</math> chooses a fiber of the tautological line bundle on <math>\mathbf{P}^r</math>, so choosing <math>r+1</math> non-simultaneously vanishing global sections of <math>L</math> determines a map from <math>X</math> into projective space <math>\mathbf{P}^r</math>. This map sends the fibers of <math>L</math> to the fibers of the dual of the tautological bundle. More specifically, suppose that <math>s_0,\dots,s_r</math> are global sections of <math>L</math>. In a small neighborhood <math>U</math> in <math>X</math>, these sections determine <math>k</math>-valued functions on <math>U</math> whose values depend on the choice of trivialization. However, they are determined up to ''simultaneous'' multiplication by a non-zero function, so their ratios are well-defined. That is, over a point <math>x</math>, the values <math>s_0(x),\dots , s_r(x)</math>are not well-defined because a change in trivialization will multiply them each by a non-zero constant λ. But it will multiply them by the ''same'' constant λ, so the [[homogeneous coordinates]] <math>[s_0(x): \dots :s_r(x)]</math> are well-defined as long as the sections <math>s_0,\dots ,s_r</math> do not simultaneously vanish at <math>x</math>.  Therefore, if the sections never simultaneously vanish, they determine a form <math>[s_0: \dots : s_r]</math> which gives a map from <math>X</math> to <math>\mathbf{P}^r</math>, and the pullback of the dual of the tautological bundle under this map is <math>L</math>. In this way, projective space acquires a [[universal property]].

The universal way to determine a map to projective space is to map to the projectivization of the vector space of all sections of <math>L</math>. In the topological case, there is a non-vanishing section at every point which can be constructed using a bump function which vanishes outside a small neighborhood of the point. Because of this, the resulting map is defined everywhere. However, the codomain is usually far, far too big to be useful. The opposite is true in the algebraic and holomorphic settings. Here the space of global sections is often finite dimensional, but there may not be any non-vanishing global sections at a given point. (As in the case when this procedure constructs a [[Lefschetz pencil]].) In fact, it is possible for a bundle to have no non-zero global sections at all; this is the case for the tautological line bundle. When the line bundle is sufficiently ample this construction verifies the [[Kodaira embedding theorem]].