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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 (), and quantities and thei ...

algebraic geometry
, 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''''X''). Linear system of dimension 1, 2, or 3 are called a
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 a
rational 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 ...

rational function
on a general variety X, or in other words of a function f in the function field of X, f \in k(X), divisors D,E \in \text(X) are linearly equivalent divisors if :D = E + (f)\ 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 \in \text(X). It is denoted , D, . Let \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 (\Gamma(X,\mathcal) \smallsetminus \)/k^\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 \mathfrak is then a projective subspace of a complete linear system, so it corresponds to a vector subspace ''W'' of \Gamma(X,\mathcal). The dimension of the linear system \mathfrak is its dimension as a projective space. Hence \dim \mathfrak = \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 \mathcal(2) on \mathbb^3 whose sections s \in \Gamma(\mathbb^3,\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 \in \Gamma(\mathbb^3,\mathcal(2)) using the rational function \left(t/s\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 + \left( \frac \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, = \mathbb(H^0(C,\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 \omega_C.


Hyperelliptic curves

One application of linear systems is used in the classification of algebraic curves. A
hyperelliptic 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 (), ...

hyperelliptic curve
is a curve C with a degree 2 morphism f:C \to \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 \mathbb^1 = \mathbb(H^0(C,\omega_C)).


grd

A g_r^d is a linear system \mathfrak on a curve C which is of degree d and dimension r. For example, hyperelliptic curves have a g^1_2 since , K_C, defines one. In fact, hyperelliptic curves have a unique g^1_2 from proposition 5.3. Another close set of examples are curves with a g_1^3 which are called trigonal curves. In fact, any curve has a g^d_1 for d \geq (1/2)g + 1.


Linear systems of hypersurfaces in \mathbb^n

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


Linear system of conics


Other examples

The
Cayley–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 of
birational 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 a
variety 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 : \operatorname(, D, ) := \bigcap_ \operatorname D_\text \ where \operatorname denotes the support of a divisor, and the intersection is taken over all effective divisors D_\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 \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 \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, \cdot C \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 \mathcal(D) on X. From this viewpoint, the base locus \operatorname(, D, ) is the set of common zeroes of all sections of \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 the
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 ...
p:\mathfrak \to \mathbb^1 given by two generic sections f,g \in \Gamma(\mathbb^n,\mathcal(d)), so \mathfrak given by the scheme
\mathfrak =\text\left( \frac \right)
This has an associated linear system of divisors since each polynomial, s_0f + t_0g for a fixed _0:t_0\in \mathbb^1 is a divisor in \mathbb^n. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of f,g, so
\text(\mathfrak) = \text\left( \frac \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 \subset \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 \otimes_k \mathcal_X \to L is surjective (here, ''k'' = the base field). Or equivalently, \operatorname((V \otimes_k \mathcal_X) \otimes_ L^) \to \bigoplus_^ \mathcal_X is surjective. Hence, writing V_X = V \times X for the trivial vector bundle and passing the surjection to the
relative 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 \hookrightarrow \mathbb(V_X^* \otimes L) \simeq \mathbb(V_X^*) = \mathbb(V^*) \times X where \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 \to \mathbb(V^*). When the base locus of ''V'' is not empty, the above discussion still goes through with \mathcal_X in the direct sum replaced by an ideal sheaf defining the base locus and ''X'' replaced by the blow-up \widetilde of it along the (scheme-theoretic) base locus ''B''. Precisely, as above, there is a surjection \operatorname((V \otimes_k \mathcal_X) \otimes_ L^) \to \bigoplus_^ \mathcal^n where \mathcal is the ideal sheaf of ''B'' and that gives rise to :i: \widetilde \hookrightarrow \mathbb(V^*) \times X. Since X - B \simeq an open subset of \widetilde, there results in the map: :f: X - B \to \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 \hookrightarrow X of algebraic varieties there is a pullback of a linear system \mathfrak on X to Y, defined as f^(\mathfrak) = \ (page 158).


O(1) on a projective variety

A projective variety X embedded in \mathbb^r has a canonical linear system determining a map to projective space from \mathcal_X(1) = \mathcal_X \otimes_ \mathcal_(1). This sends a point x \in X to its corresponding point [x_0:\cdots:x_r] \in \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