In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety ${\displaystyle X}$ amounts to understanding the different ways of mapping ${\displaystyle X}$ into projective spaces. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor.

In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bundle on a complete variety ${\displaystyle X}$ is very ample if it has enough sections to give a closed immersion (or "embedding") of ${\displaystyle X}$ into a projective space. A line bundle is ample if some positive power is very ample.

An ample line bundle on a projective variety ${\displaystyle X}$ has positive degree on every curve in ${\displaystyle X}$. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness.

Given a morphism ${\displaystyle f\colon X\to Y}$ of schemes, a vector bundle ${\displaystyle p\colon E\to Y}$ (or more generally a coherent sheaf on ${\displaystyle Y}$) has a pullback to ${\displaystyle X}$, ${\displaystyle f^{*}E=\{(x,e)\in X\times E,\;f(x)=p(e)\}}$ where the projection ${\displaystyle p'\colon f^{*}E\to X}$ is the projection on the first coordinate (see Sheaf of modules#Operations). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber of ${\displaystyle f^{*}E}$ at a point ${\displaystyle x\in X}$ is the fiber of ${\displaystyle E}$ at ${\displaystyle f(x)\in Y}$.)

The notions described in this article are related to this construction in the case of a morphism to projective space

with ${\displaystyle E={\mathcal {O}}(1)}$ the line bundle on projective space whose global sections are the homogeneous polynomials of degree 1 (that is, linear functions) in variables ${\displaystyle x_{0},\ldots ,x_{n}}$. The line bundle ${\displaystyle {\mathcal {O}}(1)}$ can also be described as the line bundle associated to a hyperplane in ${\displaystyle \mathbb {P} ^{n}}$ (because the zero set of a section of ${\displaystyle {\mathcal {O}}(1)}$ is a hyperplane). If ${\displaystyle f}$ is a closed immersion, for example, it follows that the pullback ${\displaystyle f^{*}O(1)}$ is the line bundle on ${\displaystyle X}$ associated to a hyperplane section (the intersection of ${\displaystyle X}$ with a hyperplane in ${\displaystyle \mathbb {P} ^{n}}$).

Let ${\displaystyle X}$ be a scheme over a field ${\displaystyle k}$ (for example, an algebraic variety) with a line bundle ${\displaystyle L}$. (A line bundle may also be called an invertible sheaf.) Let ${\displaystyle a_{0},...,a_{n}}$ be elements of the ${\displaystyle k}$-vector space ${\displaystyle H^{0}(X,L)}$ of global sections of ${\displaystyle L}$. The zero set of each section is a closed subset of ${\displaystyle X}$; let ${\displaystyle U}$ be the open subset of points at which at least one of ${\displaystyle a_{0},\ldots ,a_{n}}$ is not zero. Then these sections define a morphism

In more detail: for each point ${\displaystyle x}$ of ${\displaystyle U}$, the fiber of ${\displaystyle L}$ over ${\displaystyle x}$ is a 1-dimensional vector space over the residue field ${\displaystyle k(x)}$. Choosing a basis for this fiber makes ${\displaystyle a_{0}(x),\ldots ,a_{n}(x)}$ into a sequence of ${\displaystyle n+1}$ numbers, not all zero, and hence a point in projective space. Changing the choice of basis scales all the numbers by the same nonzero constant, and so the point in projective space is independent of the choice.