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| In [[algebraic geometry]], '''divisors''' are a generalization of [[codimension]] one subvarieties of [[algebraic varieties]]; two different generalizations are in common use, Cartier divisors and Weil divisors (named for [[Pierre Cartier (mathematician)|Pierre Cartier]] and [[André Weil]]). These concepts agree on [[non-singular]] varieties.
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| ==Divisors in a Riemann surface==
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| A Riemann surface is a 1-dimensional complex manifold, so its codimension 1 submanifolds are 0-dimensional. The divisors of a Riemann surface are the elements of the [[free abelian group]] over the points of the surface.
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| Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients. The '''degree''' of a divisor is the sum of its coefficients.
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| We define the divisor of a [[meromorphic function]] ''f'' as
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| :<math>(f):=\sum_{z_\nu \in R(f)} s_\nu z_\nu</math>
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| where ''R''(''f'') is the set of all zeroes and poles of ''f'', and ''s<sub>ν</sub>'' is given by
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| :<math>s_\nu := \left\{ \begin{array}{rl} a & \ \text{if } z_\nu \text{ is a zero of order }a \\
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| -a & \ \text{if } z_\nu \text{ is a pole of order }a. \end{array} \right. </math>
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| A divisor that is the divisor of a meromorphic function is called '''principal'''. It follows from the fact that a meromorphic function has as many poles as zeroes, that the degree of a principal divisor is 0. Since the divisor of a product is the sum of the divisors, the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called [[linearly equivalent]].
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| We define the divisor of a meromorphic [[1-form]] similarly. Since the space of meromorphic [[1-forms]] is a 1-dimensional vector space over the field of meromorphic functions, any two meromorphic 1-forms yield linearly equivalent divisors. The class of equivalence of these divisors is called the [[canonical divisor]] (usually denoted ''K'').
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| The [[Riemann–Roch theorem]] is an important relation between the divisors of a Riemann surface and its topology.
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| ==Weil divisor==<!-- Riemann–Roch theorem links here -->
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| A '''Weil divisor''' is a locally finite [[linear combination]] (with [[integer|integral]] coefficients) of irreducible subvarieties of [[codimension]] one. The set of Weil divisors forms an [[abelian group]] under addition. In the classical theory, where ''locally finite'' is automatic, the group of Weil divisors on a variety of [[dimension of a variety|dimension]] ''n'' is therefore the [[free abelian group]] on the (irreducible) subvarieties of dimension (''n'' − 1). For example, a '''divisor on an algebraic curve''' is a [[formal sum]] of its closed points. An '''effective Weil divisor''' is then one in which all the coefficients of the formal sum are non-negative.
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| ==Cartier divisor==
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| A '''Cartier divisor''' can be represented by an [[open cover]] by affine sets <math>{U_i}</math>, and a collection of [[rational function]]s <math>f_i</math> defined on <math>U_i</math>. The functions must be compatible in this sense: on the intersection of two sets in the cover, the quotient of the corresponding rational functions should be regular and invertible. A Cartier divisor is said to be '''effective''' if these <math>f_i</math> can be chosen to be [[regular function]]s, and in this case the Cartier divisor defines an ''associated subvariety'' of codimension 1 by forming the ideal sheaf generated locally by the <math>f_i</math>.
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| The notion can be described more conceptually with the [[function field (scheme theory)|function field]]. For each affine open subset ''U'', define ''M''′(''U'') to be the [[total quotient ring]] of ''O''<sub>''X''</sub>(''U''). Because the affine open subsets form a basis for the topology on ''X'', this defines a presheaf on ''X''. (This is not the same as taking the total quotient ring of ''O''<sub>''X''</sub>(''U'') for arbitrary ''U'', since that does not define a presheaf.<ref>Kleiman, p. 203</ref>) The sheaf ''M''<sub>''X''</sub> of rational functions on ''X'' is the sheaf associated to the presheaf ''M''′, and the quotient sheaf {{nowrap|''M''<sub>''X''</sub><sup>*</sup> / ''O''<sub>''X''</sub><sup>*</sup>}} is the sheaf of local Cartier divisors.
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| A Cartier divisor is a global section of the quotient sheaf ''M''<sub>''X''</sub><sup>*</sup>/''O''<sub>''X''</sub><sup>*</sup>. We have the exact sequence <math>1 \to \mathcal O^*_X \to M^*_X \to M^*_X / \mathcal O^*_X \to 1</math>, so, applying the global section functor <math>\Gamma (X, \bullet)</math> gives the exact sequence <math>1 \to \Gamma (X, O^*_X) \to \Gamma (X, M^*_X) \to \Gamma (X, M^*_X / \mathcal O^*_X) \to H^1(X, \mathcal O^*_X)</math>.
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| A Cartier divisor is said to be principal if it is in the range of the morphism <math>\Gamma (X, M^*_X) \to \Gamma (X, M^*_X / \mathcal O^*_X)</math>, that is, if it is the class of a global rational function.
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| ===Cartier divisors in nonrigid sheaves===
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| Of course the notion of Cartier divisors exists in any sheaf. But if the sheaf is not rigid enough, the notion tends to lose some of its interest. For example in a [[fine sheaf]] (e.g. the sheaf of real-valued continuous, or smooth, functions on an open subset of a [[euclidean space]], or locally homeomorphic, or diffeomorphic, to such a set, such as a [[topological manifold]]), any local section is a divisor of 0, so that the total quotient sheaves are zero, so that the sheaf contains no non-trivial Cartier divisor.
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| ==From Cartier divisors to Weil divisor==
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| There is a natural homomorphism from the group of Cartier divisors to that of Weil divisors, which is an isomorphism for integral separated Noetherian schemes provided that all local rings are unique factorization domains.
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| In general Cartier ''behave better'' than Weil divisors when the variety has [[Mathematical singularity|singular points]].
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| An example of a surface on which the two concepts differ is a ''cone'', i.e. a singular [[quadric]]. At the (unique) singular point, the vertex of the cone, a single line drawn on the cone is a Weil divisor — but is not a Cartier divisor.
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| The ''divisor'' appellation is part of the history of the subject, going back to the [[Richard Dedekind|Dedekind]]–[[Heinrich M. Weber|Weber]] work which in effect showed the relevance of [[Dedekind domain]]s to the case of [[algebraic curve]]s.<ref>Section VI.6 of {{harvtxt|Dieudonné|1985}}.</ref> In that case the free abelian group on the points of the curve is closely related to the [[fractional ideal]] theory.
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| ==From Cartier divisors to line bundles==
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| The notion of [[Trivialization_(mathematics)#Structure_groups_and_transition_functions|transition map]] associates naturally to every Cartier divisor ''D'' a [[line bundle]] (strictly, [[invertible sheaf]]) commonly denoted by ''O''<sub>''X''</sub>(''D'').
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| The line bundle <math>\mathcal L (D)</math> associated to the Cartier divisor ''D'' is the sub-bundle of the sheaf ''M''<sub>''X''</sub> of rational fractions described above whose stalk at <math>x \in X</math> is given by <math>D_x \in \Gamma (x, M^*_X/\mathcal O^*_X)</math> viewed as a line on the stalk at ''x'' of <math>\mathcal O_X</math> in the stalk at ''x'' of <math>M_X</math>. The subsheaf thus described is tautologically locally freely monogenous over the structure sheaf <math>\mathcal O_X</math>.
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| The mapping <math>D \mapsto \mathcal L (D)</math> is a group homomorphism: the sum of divisors corresponds to the [[tensor product]] of line bundles, and isomorphism of bundles corresponds precisely to [[linear equivalence]] of Cartier divisors. The group of divisors classes modulo linear equivalence therefore [[injective function|injects]] into the [[Picard group]]. The mapping is not surjective for all compact complex manifolds, but surjectivity does hold
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| for all smooth projective varieties. The latter is true because, by the Kodaira embedding theorem, the tensor product of any line bundle with a sufficiently high power of any positive line bundle becomes ample; thus, on any such manifold, any line bundle is the formal difference between
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| two ample line bundles, and any ample line bundle may be viewed as an effective divisor.
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| ==Global sections of line bundles and linear systems==
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| Recall that the local equations of a Cartier divisor <math>D</math> in a variety <math>X</math> give rise to transition maps for a line bundle <math> \mathcal L (D)</math>, and linear equivalences induce isomorphism of line bundles.
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| Loosely speaking, a Cartier divisor ''D'' is said to be ''effective'' if it is the zero locus of a global section of its associated line bundle <math>\mathcal L(D)</math>. In terms of the definition above, this means that its local equations coincide with the equations of the vanishing locus of a global section.
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| From the divisor linear equivalence/line bundle isormorphism principle, a Cartier divisor is linearly equivalent to an effective divisor if, and only if, its associate line bundle <math>\mathcal L (D)</math> has non-zero global sections. Two collinear non-zero global sections have the same vanishing locus, and hence the projective space <math>\mathbb P \Gamma (X, \mathcal L (D))</math> over '''k''' identifies with the set of effective divisors linearly equivalent to <math>D</math>.
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| If <math>X</math> is a [[projective variety|projective]] (or [[complete variety|proper]]) variety over a field <math>k</math>, then <math>\Gamma (X, \mathcal L (D))</math> is a finite dimensional <math>k</math>-vector space, and the associated projective space over <math>k</math> is called the complete linear system of <math>D</math>. Its linear subspaces are called [[linear systems of divisors]]. The [[Riemann-Roch theorem for algebraic curves]] is a fundamental identity involving the dimension of complete linear systems in the setup of projective curves.
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| ==See also==
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| * [[ample divisor]]
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| * [[Arakelov divisor]]
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| * [[nef divisor]]
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| * [[Theta-divisor]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{Citation
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| | last=Dieudonné
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| | first=Jean
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| | author-link=Jean Dieudonné
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| | title=History of algebraic geometry. An outline of the history and development of algebraic geometry (Translated from the French by Judith D. Sally)
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| | publisher=Wadsworth International Group
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| | location=Belmont, CA
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| | series=Wadsworth Mathematics Series
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| | isbn=0-534-03723-2
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| | mr=0780183
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| | year=1985}}
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| * Section II.6 of {{Citation
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| | last=Hartshorne
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| | first=Robin
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| | author-link=Robin Hartshorne
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| | title=Algebraic Geometry
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| | publisher=
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| | location=New York, Heidelberg
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| | series=Graduate Texts in Mathematics
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| | isbn=0-387-90244-9
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| | mr=0463157
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| | year=1977
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| | volume=52
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| }}
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| *{{citation |last=Kleiman |first=Steven |title=Misconceptions about ''K''<sub>''X''</sub> |journal=L'Enseignment Mathématique |volume=25 |year=1979 |pages=203–206 |doi=10.5169/seals-50379}}
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| {{DEFAULTSORT:Divisor (Algebraic Geometry)}}
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| [[Category:Geometry of divisors]]
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My name is Roseanne Sizer but everybody calls me Roseanne. I'm from Italy. I'm studying at the college (3rd year) and I play the Tuba for 8 years. Usually I choose songs from the famous films ;).
I have two brothers. I like Footbag, watching TV (Grey's Anatomy) and Target Shooting.
my web site - FIFA coin Generator