In _{''X''}).
Linear system of dimension 1, 2, or 3 are called a

algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
These arose first in the form of a ''linear system'' of algebraic curve
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s in the projective plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s ''D'' on a general scheme or even a ringed spaceIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

(''X'', ''O''pencil
A pencil is a writing
Writing is a medium of human communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was or ...

, a net, or a web, respectively.
A map determined by a linear system is sometimes called the Kodaira map.
Definition

Given the fundamental idea of arational function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

on a general variety $X$, or in other words of a function $f$ in the function field of $X$, $f\; \backslash in\; k(X)$, divisors $D,E\; \backslash in\; \backslash text(X)$ are linearly equivalent divisors if
:$D\; =\; E\; +\; (f)\backslash $
where $(f)$ denotes the divisor of zeroes and poles of the function $f$.
Note that if $X$ has singular points, 'divisor' is inherently ambiguous (Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil
André W ...

s, Weil divisors: see divisor (algebraic geometry)
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commuta ...

). The definition in that case is usually said with greater care (using invertible sheaves
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...

or holomorphic line bundleIn mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is Holomorphic function, holomorphic. Fundamental examples are the holomorphic ta ...

s); see below.
A complete linear system on $X$ is defined as the set of all effective divisors linearly equivalent to some given divisor $D\; \backslash in\; \backslash text(X)$. It is denoted $,\; D,$. Let $\backslash mathcal$ be the line bundle associated to $D$. In the case that $X$ is a nonsingular projective variety the set $,\; D,$ is in natural bijection with $(\backslash Gamma(X,\backslash mathcal)\; \backslash smallsetminus\; \backslash )/k^\backslash ast,$ 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 and is therefore a projective space.
A linear system $\backslash mathfrak$ is then a projective subspace of a complete linear system, so it corresponds to a vector subspace ''W'' of $\backslash Gamma(X,\backslash mathcal).$ The dimension of the linear system $\backslash mathfrak$ is its dimension as a projective space. Hence $\backslash dim\; \backslash mathfrak\; =\; \backslash dim\; W\; -\; 1$.
Since a Cartier divisor class is an isomorphism class of a line bundle, linear systems can also be introduced by means of the line bundleIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

or invertible sheaf
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

language, without reference to divisors at all. In those terms, divisors $D$ (Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil
André W ...

s, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.
Examples

Linear equivalence

Consider the line bundle $\backslash mathcal(2)$ on $\backslash mathbb^3$ whose sections $s\; \backslash in\; \backslash Gamma(\backslash mathbb^3,\backslash mathcal(2))$ define quadric surfaces. For the associated divisor $D\_s\; =\; Z(s)$, it is linearly equivalent to any other divisor defined by the vanishing locus of some $t\; \backslash in\; \backslash Gamma(\backslash mathbb^3,\backslash mathcal(2))$ using the rational function $\backslash left(t/s\backslash right)$ (Proposition 7.2). For example, the divisor $D$ associated to the vanishing locus of $x^2\; +\; y^2\; +\; z^2\; +\; w^2$ is linearly equivalent to the divisor $E$ associated to the vanishing locus of $xy$. Then, there is the equivalence of divisors$D\; =\; E\; +\; \backslash left(\; \backslash frac\; \backslash right)$

Linear systems on curves

One of the important complete linear systems on an algebraic curve $C$ of genus $g$ is given by the complete linear system associated with the canonical divisor $K$, denoted $,\; K,\; =\; \backslash mathbb(H^0(C,\backslash omega\_C))$. This definition follows from proposition II.7.7 of Hartshorne since every effective divisor in the linear system comes from the zeros of some section of $\backslash omega\_C$.Hyperelliptic curves

One application of linear systems is used in the classification of algebraic curves. Ahyperelliptic curve
In algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), ...

is a curve $C$ with a degree $2$ morphism $f:C\; \backslash to\; \backslash mathbb^1$. For the case $g=2$ all curves are hyperelliptic: the Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which ...

then gives the degree of $K\_C$ is $2g\; -\; 2\; =\; 2$ and $h^0(K\_C)\; =\; 2$, hence there is a degree $2$ map to $\backslash mathbb^1\; =\; \backslash mathbb(H^0(C,\backslash omega\_C))$.
g_{r}^{d}

Linear systems of hypersurfaces in $\backslash mathbb^n$

Consider the line bundle $\backslash mathcal(d)$ over $\backslash mathbb^n$. If we take global sections $V\; =\; \backslash Gamma(\backslash mathcal(d))$, then we can take its projectivization $\backslash mathbb(V)$. This is isomorphic to $\backslash mathbb^N$ where :$N\; =\; \backslash binom\; -\; 1$ Then, using any embedding $\backslash mathbb^k\; \backslash to\; \backslash mathbb^N$ we can construct a linear system of dimension $k$.Linear system of conics

Other examples

TheCayley–Bacharach theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.
Linear systems in birational geometry

In general linear systems became a basic tool ofbirational geometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

as practised by the Italian school of algebraic geometry
In relation to the history of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the world ...

. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions — the Riemann–Roch problem as it can be called — can be better phrased in terms of homological algebra
Homological algebra is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contain ...

. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...

generated by codimension-one subvarieties), and Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil
André W ...

s coming from sections of invertible sheaves
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...

.
The Italian school liked to reduce the geometry on an algebraic surface
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

to that of linear systems cut out by surfaces in three-space; Zariski
, birth_date =
, birth_place = Kobrin
Kobryn ( be, Кобрын; russian: Кобрин; pl, Kobryń; lt, Kobrynas; uk, Кобринь, Kobryn'; yi, קאָברין) is a city in the Brest Region of Belarus and the center of th ...

wrote his celebrated book ''Algebraic Surfaces'' to try to pull together the methods, involving ''linear systems with fixed base points''. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the s ...

's characteristic linear system of an algebraic family of curves on an algebraic surface.
Base locus

The base locus of a linear system of divisors on avariety
Variety may refer to:
Science and technology
Mathematics
* Algebraic variety, the set of solutions of a system of polynomial equations
* Variety (universal algebra), classes of algebraic structures defined by equations in universal algebra
Hort ...

refers to the subvariety of points 'common' to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines $x=a$ has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.
More precisely, suppose that $,\; D,$ is a complete linear system of divisors on some variety $X$. Consider the intersection
: $\backslash operatorname(,\; D,\; )\; :=\; \backslash bigcap\_\; \backslash operatorname\; D\_\backslash text\; \backslash $
where $\backslash operatorname$ denotes the support of a divisor, and the intersection is taken over all effective divisors $D\_\backslash text$ in the linear system. This is the base locus of $,\; D,$ (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheafIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of $\backslash operatorname$ should be).
One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose $,\; D,$ is such a class on a variety $X$, and $C$ an irreducible curve on $X$. If $C$ is not contained in the base locus of $,\; D,$, then there exists some divisor $\backslash tilde\; D$ in the class which does not contain $C$, and so intersects it properly. Basic facts from intersection theory then tell us that we must have $,\; D,\; \backslash cdot\; C\; \backslash geq\; 0$. The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef.
In the modern formulation of algebraic geometry, a complete linear system $,\; D,$ of (Cartier) divisors on a variety $X$ is viewed as a line bundle $\backslash mathcal(D)$ on $X$. From this viewpoint, the base locus $\backslash operatorname(,\; D,\; )$ is the set of common zeroes of all sections of $\backslash mathcal(D)$. A simple consequence is that the bundle is globally generated if and only if the base locus is empty.
The notion of the base locus still makes sense for a non-complete linear system as well: the base locus of it is still the intersection of the supports of all the effective divisors in the system.
Example

Consider theLefschetz pencilIn mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety ''V''.
Description
A ''pencil'' is a particular kind of linear system of di ...

$p:\backslash mathfrak\; \backslash to\; \backslash mathbb^1$ given by two generic sections $f,g\; \backslash in\; \backslash Gamma(\backslash mathbb^n,\backslash mathcal(d))$, so $\backslash mathfrak$ given by the scheme$\backslash mathfrak\; =\backslash text\backslash left(\; \backslash frac\; \backslash right)$This has an associated linear system of divisors since each polynomial, $s\_0f\; +\; t\_0g$ for a fixed $;\; href="/html/ALL/s/\_0:t\_0.html"\; ;"title="\_0:t\_0">\_0:t\_0$ is a divisor in $\backslash mathbb^n$. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of $f,g$, so

$\backslash text(\backslash mathfrak)\; =\; \backslash text\backslash left(\; \backslash frac\; \backslash right)$

A map determined by a linear system

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; see the section below) Let ''L'' be a line bundle on an algebraic variety ''X'' and $V\; \backslash subset\; \backslash Gamma(X,\; L)$ 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 $V\; \backslash otimes\_k\; \backslash mathcal\_X\; \backslash to\; L$ is surjective (here, ''k'' = the base field). Or equivalently, $\backslash operatorname((V\; \backslash otimes\_k\; \backslash mathcal\_X)\; \backslash otimes\_\; L^)\; \backslash to\; \backslash bigoplus\_^\; \backslash mathcal\_X$ is surjective. Hence, writing $V\_X\; =\; V\; \backslash times\; X$ for the trivial vector bundle and passing the surjection to therelative Proj
In algebraic geometry, Proj is a construction analogous to the spectrum of a ring, spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective variety, projective varieti ...

, there is a closed immersion:
:$i:\; X\; \backslash hookrightarrow\; \backslash mathbb(V\_X^*\; \backslash otimes\; L)\; \backslash simeq\; \backslash mathbb(V\_X^*)\; =\; \backslash mathbb(V^*)\; \backslash times\; X$
where $\backslash simeq$ on the right is the invariance of the projective bundle under a twist by a line bundle. Following ''i'' by a projection, there results in the map:
:$f:\; X\; \backslash to\; \backslash mathbb(V^*).$
When the base locus of ''V'' is not empty, the above discussion still goes through with $\backslash mathcal\_X$ in the direct sum replaced by an ideal sheaf defining the base locus and ''X'' replaced by the blow-up $\backslash widetilde$ of it along the (scheme-theoretic) base locus ''B''. Precisely, as above, there is a surjection $\backslash operatorname((V\; \backslash otimes\_k\; \backslash mathcal\_X)\; \backslash otimes\_\; L^)\; \backslash to\; \backslash bigoplus\_^\; \backslash mathcal^n$ where $\backslash mathcal$ is the ideal sheaf of ''B'' and that gives rise to
:$i:\; \backslash widetilde\; \backslash hookrightarrow\; \backslash mathbb(V^*)\; \backslash times\; X.$
Since $X\; -\; B\; \backslash simeq$ an open subset of $\backslash widetilde$, there results in the map:
:$f:\; X\; -\; B\; \backslash to\; \backslash mathbb(V^*).$
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).
Linear system determined by a map to a projective space

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. For a closed immersion $f:\; Y\; \backslash hookrightarrow\; X$ of algebraic varieties there is a pullback of a linear system $\backslash mathfrak$ on $X$ to $Y$, defined as $f^(\backslash mathfrak)\; =\; \backslash $ (page 158).O(1) on a projective variety

A projective variety $X$ embedded in $\backslash mathbb^r$ has a canonical linear system determining a map to projective space from $\backslash mathcal\_X(1)\; =\; \backslash mathcal\_X\; \backslash otimes\_\; \backslash mathcal\_(1)$. This sends a point $x\; \backslash in\; X$ to its corresponding point $[x\_0:\backslash cdots:x\_r]\; \backslash in\; \backslash mathbb^r$.See also

* Brill–Noether theory *Lefschetz pencilIn mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety ''V''.
Description
A ''pencil'' is a particular kind of linear system of di ...

*bundle of principal parts
References

* * Hartshorne, R. ''Algebraic Geometry'', Springer-Verlag, 1977; corrected 6th printing, 1993. . * Lazarsfeld, R., ''Positivity in Algebraic Geometry I'', Springer-Verlag, 2004. . {{refend Geometry of divisors