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Normal Cone
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry. Definition The normal cone or C_ of an embedding , defined by some sheaf of ideals ''I'' is defined as the relative Spec \operatorname_X \left(\bigoplus_^ I^n / I^\right). When the embedding ''i'' is regular embedding, regular the normal cone is the normal bundle, the vector bundle on ''X'' corresponding to the dual of the sheaf . If ''X'' is a point, then the normal cone and the normal bundle to it are also called the tangent cone and the tangent space (Zariski tangent space) to the point. When ''Y'' = Spec ''R'' is affine, the definition means that the normal cone to ''X'' = Spec ''R''/''I'' is the Spec of the associated graded ring of ''R'' with respect to ''I''. If ''Y'' is the product ''X'' × ''X'' and the embedding ''i'' is the diagonal embedding, then the normal bundle to ''X'' in ''Y'' is the tangent bundle to ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 manifold, and S \subset M a Riemannian submanifold. Define, for a given p \in S, a vector n \in \mathrm_p M to be '' normal'' to S whenever g(n,v)=0 for all v\in \mathrm_p S (so that n is orthogonal to \mathrm_p S). The set \mathrm_p S of all such n is then called the ''normal space'' to S at p. Just as the total space of the tangent bundle to a manifold is constructed from all tangent spaces to the manifold, the total space of the normal bundle \mathrm S to S is defined as :\mathrmS := \coprod_ \mathrm_p S. The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle. General definition More abstractly, given an immersion i: N \to M (for instance an em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Dimension
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residual Intersection
In algebraic geometry, the problem of residual intersection asks the following: :''Given a subset ''Z'' in the intersection \bigcap_^r X_i of varieties, understand the complement of ''Z'' in the intersection; i.e., the residual set to ''Z''.'' The intersection determines a class (X_1 \cdots X_r), the intersection product, in the Chow group of an ambient space and, in this situation, the problem is to understand the class, the residual class to ''Z'': :(X_1 \cdots X_r) - (X_1 \cdots X_r)^Z where -^Z means the part supported on ''Z''; classically the degree of the part supported on ''Z'' is called the equivalence of ''Z''. The two principal applications are the solutions to problems in enumerative geometry (e.g., Steiner's conic problem) and the derivation of the multiple-point formula, the formula allowing one to count or enumerate the points in a fiber even when they are infinitesimally close. The problem of residual intersection goes back to the 19th century. The modern formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segre Class
In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Beniamino Segre (1953). In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton (1998), Segre classes play a fundamental role. Definition Suppose C is a cone over X , q is the projection from the projective completion \mathbb(C \oplus 1) of C to X, and \mathcal(1) is the anti-tautological line bundle on \mathbb(C \oplus 1). Viewing the Chern class c_1(\mathcal(1)) as a group endomorphism of the Chow group of \mathbb(C \oplus 1), the total Segre class of C is given by: :s(C) = q_* \left( \sum_ c_1(\mathcal(1)) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Cone
Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group where the commutator subgroup is abelian * Abelianisation Topology and number theory * Abelian variety, a complex torus that can be embedded into projective space * Abelian surface, a two-dimensional abelian variety * Abelian function, a meromorphic function on an abelian variety * Abelian integral, a function related to the indefinite integral of a differential of the first kind Other mathematics * Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel * Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series * Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Stack
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks. Definition A quotient stack is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' an ''S''-scheme on which ''G'' acts. Let the quotient stack /G/math> be the category over the category of ''S''-schemes, where *an object over ''T'' is a principal ''G''-bundle P\to T together with equivariant map P\to X; *a morphism from P\to T to P'\to T' is a bundle map (i.e., forms a commutative diagram) that is comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Stack
In algebraic geometry, given algebraic stacks p: X \to C, \, q: Y \to C over a base category ''C'', a morphism f: X \to Y of algebraic stacks is a functor such that q \circ f = p. More generally, one can also consider a morphism between prestacks (a stackification would be an example). Types One particular important example is a presentation of a stack, which is widely used in the study of stacks. An algebraic stack ''X'' is said to be smooth of dimension ''n'' - ''j'' if there is a smooth presentation U \to X of relative dimension ''j'' for some smooth scheme ''U'' of dimension ''n''. For example, if \operatorname_n denotes the moduli stack of rank-''n'' vector bundles, then there is a presentation \operatorname(k) \to \operatorname_n given by the trivial bundle \mathbb^n_k over \operatorname(k). A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.§ 8.6 of F. MeyerNotes on algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsor (algebraic Geometry)
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra. Though other notions of torsors are known in more general context (e.g. over stacks) this article will focus on torsors over schemes, the original setting where torsors have been thought for. The word ''torsor'' comes from the French ''torseur''. They are indeed widely discussed, for instance, in Michel Demazure's and Pierre Gabriel's famous book ''Groupes algébriques, Tome I''. Definition Let \mathcal be a Grothendieck topology and X a scheme. Moreover let G be a group scheme over X, a G-torsor (or principal G-bundle) over X for the topology \mathcal (or simply a G-torsor when the topology is clear from the context) is the data of a scheme P and a morphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Stack
In algebraic geometry, a quotient stack is a stack (mathematics), stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a Scheme (mathematics), scheme or a algebraic variety, variety by a Group (mathematics), group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks. Definition A quotient stack is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' an ''S''-scheme on which ''G'' group-scheme action, acts. Let the quotient stack [X/G] be the fibered category, category over the category of ''S''-schemes, where *an object over ''T'' is a torsor (algebraic geometry), principal ''G''-bundle P\to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic objects, the corresponding cotangent complex \mathbf_^\bullet can be thought of as a universal "linearization" of it, which serves to control the deformation theory of f. It is constructed as an object in a certain derived category of sheaves on X using the methods of homotopical algebra. Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne–Mumford Stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened to " smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford. A key fact about a Deligne–Mumford stack ''F'' is that any ''X'' in F(B), where ''B'' is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...; see groupoid scheme. Examples Affine Stacks Deligne–Mumford stacks are typica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regularly Embedded
In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension ''r'' if each point ''x'' in ''X'' has an open affine neighborhood ''U'' in ''Y'' such that the ideal of X \cap U is generated by a regular sequence of length ''r''. A regular embedding of codimension one is precisely an effective Cartier divisor. Examples and usage For example, if ''X'' and ''Y'' are smooth morphism, smooth over a scheme ''S'' and if ''i'' is an ''S''-morphism, then ''i'' is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If \operatornameB is regularly embedded into a regular scheme, then ''B'' is a complete intersection ring. The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when ''i'' is a regular embedding, if ''I'' is the ideal sheaf of ''X'' in ''Y'', then the normal sheaf, the dual of I/I^2, is locally free (thus a vector b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |