characteristic linear system of an algebraic family of curves
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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, 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 curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors ''D'' on a general
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
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'', ''O''''X''). Linear system of dimension 1, 2, or 3 are called a pencil, 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, 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 rat ...
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 by David Mu ...
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 (mathematician), Pierre Cartier ...
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 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 bundle In 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. Fundamental examples are the holomorphic tangent bundle of a com ...
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 elements of the set , D, , which can be written as E = (f)+D , are 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 E = (f)+D to f)/math> (this is well defined since (\lambda f) = \lambda (f)) 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 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 organisin ...
or invertible sheaf 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 by David Mu ...
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 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 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 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 m ...
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 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 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 (mathematician), Pierre Cartier ...
s (in the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
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 Mu ...
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, 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 ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
'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 r ...
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 In 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 d ...
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 construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not fun ...
, 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 In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme ''X'' over a Noetherian scheme ''S'' is a P''n''-bundle if it is locally a projective ''n''-space; i.e., X \times_S U \simeq \math ...
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 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 *
Lefschetz pencil In 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 d ...
*
bundle of principal parts In algebraic geometry, given a line bundle ''L'' on a smooth variety ''X'', the bundle of ''n''-th order principal parts of ''L'' is a vector bundle of rank \tbinom that, roughly, parametrizes ''n''-th order Taylor expansions of sections of&nbs ...


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