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Diviseur (géométrie algébrique)
Concepts associés (27)
Chow group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product.
Invertible sheaf
In mathematics, an invertible sheaf is a sheaf on a ringed space which 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 interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Let (X, OX) be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of tensor product of OX-modules. The identity element for this operation is OX itself.
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an , and so they are closed under operations such as taking , , and cokernels.
Linear system of divisors
In 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 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 or even a ringed space (X, OX).
Dualité de Serre
En géométrie algébrique, la dualité de Serre est une dualité pour la cohomologie cohérente de variétés algébriques, démontrée par Jean-Pierre Serre. La version originale s'applique aux fibrés vectoriels sur une variété projective lisse, mais Alexander Grothendieck la généralise largement. Sur une variété de dimension n, le théorème énonce l'isomorphisme d'un groupe de cohomologie avec l'espace dual d'un autre, le . La dualité de Serre est l'analogue pour la cohomologie cohérente de la dualité de Poincaré en topologie.
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available. Let R and S be commutative rings and φ : R → S be a ring homomorphism.
Minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible".
Nef line bundle
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X.
Tautological bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.
Variété abélienne
En mathématiques, et en particulier, en géométrie algébrique et géométrie complexe, une variété abélienne A est une variété algébrique projective qui est un groupe algébrique. La condition de est l'équivalent de la compacité pour les variétés différentielles ou analytiques, et donne une certaine rigidité à la structure. C'est un objet central en géométrie arithmétique. Une variété abélienne sur un corps k est un groupe algébrique A sur k, dont la variété algébrique sous-jacente est projective, connexe et géométriquement réduite.