linear system of divisors
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In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, 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, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
s in the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
s ''D'' on a general scheme or even a
ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
(X, \mathcal_X). Linear systems of dimension 1, 2, or 3 are called a
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of ...
, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map.


Definitions

Given a general variety X, two divisors D,E \in \text(X) are linearly equivalent if :E = D + (f)\ for some non-zero
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
f on X, or in other words a non-zero element f of the function field k(X). Here (f) denotes the divisor of zeroes and poles of the function f. Note that if X has singular points, the notion of '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 by David Mumf ...
s,
Weil 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 by David Mumfo ...
s: see
divisor (algebraic geometry) 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 by David Mum ...
). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); 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 by associating the element E = D + (f) of , D, to the set of non-zero multiples of f (this is well defined since two non-zero rational functions have the same divisor if and only if they are non-zero multiples of each other). A complete linear system , D, 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 . Linear systems can also be introduced by means of the
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
or
invertible sheaf In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their intera ...
language. 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 by David Mumf ...
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 surface In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hyper ...
s. 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 Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
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, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
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, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
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)).


gdr

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 which is induced by the 2:1-map C \to \mathbb P^1. 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 a projective space

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


Characteristic linear system of a family of curves

The characteristic linear system of a family of curves on an algebraic surface ''Y'' for a curve ''C'' in the family is a linear system formed by the curves in the family that are infinitely near ''C''. In modern terms, it is a subsystem of the linear system associated to the
normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian ...
to C \hookrightarrow Y. Note a characteristic system need not to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the Kodaira–Spencer theory can be used to answer the question of the completeness.


Other examples

The
Cayley–Bacharach theorem In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: :Assume that two cubics and in the projective plane meet in nine (different) poi ...
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, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying Map (mathematics), mappings that are gi ...
as practised by the Italian school of algebraic geometry. 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 that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. The effect of working on varieties with singular points is to show up a difference between
Weil 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 by David Mumfo ...
s (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 (mathematics), operation that is associative, commutative, and inverti ...
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 by David Mumf ...
s coming from sections of invertible sheaves. The Italian school liked to reduce the geometry on an
algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
to that of linear systems cut out by surfaces in three-space; Zariski 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 (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
'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 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 sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
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 pencil 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, there is a
closed immersion In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formali ...
: :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 In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal (ring theory), ideal in a ring (mathematics), ring. The ideal sheaves on a geometric object are closely connected to its sub ...
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 natural 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 _0:\cdots:x_r\in \mathbb^r .


See also

*
Brill–Noether theory In algebraic geometry, Brill–Noether theory, introduced by , is the study of special divisors, certain divisors on a curve that determine more compatible functions than would be predicted. In classical language, special divisors move on the cu ...
* Lefschetz pencil * bundle of principal parts


References

* * Hartshorne, R., ''Algebraic Geometry'',
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, 1977; corrected 6th printing, 1993. . * Lazarsfeld, R., ''Positivity in Algebraic Geometry I'', Springer-Verlag, 2004. . {{refend Geometry of divisors