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Normal Cone
In algebraic geometry, the normal cone C_XY of a subscheme X of a scheme Y 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 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 ''X''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Spec
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. Zariski topology For any ideal ''I'' of ''R'', define V_I to be the set of prime ideals containing ''I''. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of \operatorname(R), and \ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a ... [...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 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 bundle) and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry (book)
''Algebraic Geometry'' is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.MathSciNet lists more than 2500 citations of this book. Importance It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students. Contents The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ..., with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergebnisse Der Mathematik Und Ihrer Grenzgebiete
''Ergebnisse der Mathematik und ihrer Grenzgebiete''/''A Series of Modern Surveys in Mathematics'' is a series of scholarly monographs published by Springer Science+Business Media. The title literally means "Results in mathematics and related areas". Most of the books were published in German or English, but there were a few in French and Italian. There have been several sequences, or ''Folge'': the original series, neue Folge, and 3.Folge. Some of the most significant mathematical monographs of 20th century appeared in this series. Original series The series started in 1932 with publication of ''Knotentheorie'' by Kurt Reidemeister as "Band 1" (English: volume 1). There seems to have been double numeration in this sequence. Neue Folge This sequence started in 1950 with the publication of ''Transfinite Zahlen'' by Heinz Bachmann. The volumes are consecutively numbered, designated as either "Band" or "Heft". A total of 100 volumes was published, often in multiple editions, but pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Fundamental Class
In mathematics, specifically enumerative geometry, the virtual fundamental class \text_ of a space X is a replacement of the classical fundamental class \in A^*(X) in its chow ring which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree d rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces\overline_(X,\beta)for X a scheme and \beta a class in A_1(X), their behavior can be wild at the boundary, such aspg 503 having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space\overline_(\mathbb^2,1 for H the class of a line in \mathbb^2. The non-compact "smooth" component is empty, but the boundary contains maps of curvesf:C \to \mathbb^2whose components consist of one degree 3 curve which contracts to a point. There is a vir ... [...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 formul ... [...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 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))^ mathbb(C ... [...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 * A ... [...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: *an object over ''T'' is a principal ''G''-bundle P\to T together with equivariant map P\to X; *an arrow from P\to T to P'\to T' is a bundle map (i.e., forms a commutative diagram) that is compatible with ... [...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 analog 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. The category of torsors over a fixed base forms a stack. Conversely, a prestack can be stackified by taking the category of torsors (over the prestack). Definition Given a smooth algebraic group ''G'', a ''G''-torsor (or a principal ''G''-bundle) ''P'' over a scheme ''X'' is a scheme (or even algebraic space) with an action of ''G'' that is locally trivial in the given Grothendieck topology in the sense that the base change Y \times_X P along some covering map Y \to X is isomorphic to the trivial torsor Y \times G \to Y (''G'' acts only on the second factor). Equivalently, a ''G''-torsor ''P'' on ''X'' is a principal homogeneous space for the group scheme G_X = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |