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Contracting Morphism
In algebraic geometry, a contraction morphism is a surjective projective morphism f: X \to Y between normal projective varieties (or projective schemes) such that f_* \mathcal_X = \mathcal_Y or, equivalently, the geometric fibers are all connected ( Zariski's connectedness theorem). It is also commonly called an algebraic fiber space, as it is an analog of a fiber space in algebraic topology. By the Stein factorization, any surjective projective morphism is a contraction morphism followed by a finite morphism. Examples include ruled surfaces and Mori fiber spaces. Birational perspective The following perspective is crucial in birational geometry (in particular in Mori's minimal model program). Let ''X'' be a projective variety and \overline(X) the closure of the span of irreducible curves on ''X'' in N_1(X) = the real vector space of numerical equivalence classes of real 1-cycles on ''X''. Given a face ''F'' of \overline(X), the contraction morphism associated to ''F'', if it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Morphism
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] |
Zariski's Connectedness Theorem
In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected. Statement Suppose that ''f'' is a proper surjective morphism of varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ... from ''X'' to ''Y'' such that the function field of ''Y'' is separably closed in that of ''X''. Then Zariski's connectedness theorem says that the inverse image of any norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Space
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ E \to T_B is surjective, or, equivalently, its rank equals \dim B. History In topology, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1932, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which vector bun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein Factorization
In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points. Statement One version for schemes states the following: Let ''X'' be a scheme, ''S'' a locally noetherian scheme and f: X \to S a proper morphism. Then one can write :f = g \circ f' where g\colon S' \to S is a finite morphism and f'\colon X \to S' is a proper morphism so that f'_* \mathcal_X = \mathcal_. The existence of this decomposition itself is not difficult. See below. But, by Zariski's connectedness theorem, the last part in the above says that the fiber f'^(s) is connected for any s \in S. It follows: Corollary: For any s \in S, the set of connected components of the fiber f^(s) is in bijection with the set of points in the fiber g^(s). Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruled Surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is ''doubly ruled'' if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points . The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mori Fiber Space
In algebraic geometry, a Fano fibration or Fano fiber space, named after Gino Fano, is a morphism of varieties whose general fiber is a Fano variety (in other words has ample anticanonical bundle) of positive dimension. The ones arising from extremal contractions in the minimal model program are called Mori fibrations or Mori fiber spaces (for Shigefumi Mori). They appear as standard forms for varieties without a minimal model. See also * Ample line bundle * Fiber bundle * Fibration * Quasi-fibration In algebraic topology, a quasifibration is a generalisation of fibre bundles and fibrations introduced by Albrecht Dold and René Thom. Roughly speaking, it is a continuous map ''p'': ''E'' → ''B'' having the same behaviour as a fibration regardi ... References * Algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mori's 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. Outline 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". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety X for which any birational morphism f\colon X \to X' with a smooth surface X' is an isomorphism. In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety X, which for simplicity is assumed non-singular. There are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone Theorem
In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X. Definition Let X be a proper variety. By definition, a (real) ''1-cycle'' on X is a formal linear combination C=\sum a_iC_i of irreducible, reduced and proper curves C_i, with coefficients a_i \in \mathbb. ''Numerical equivalence'' of 1-cycles is defined by intersections: two 1-cycles C and C' are numerically equivalent if C \cdot D = C' \cdot D for every Cartier divisor D on X. Denote the real vector space of 1-cycles modulo numerical equivalence by N_1(X). We define the ''cone of curves'' of X to be : NE(X) = \left\ where the C_i are irreducible, reduced, proper curves on X, and _i/math> their classes in N_1(X). It is not difficult to see that NE(X) is indeed a convex cone in the sense of convex geometry. Applications One useful application of the notion of the cone of curves is the Kleiman condition, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Castelnuovo's Contraction Theorem
In mathematics, Castelnuovo's contraction theorem is used in the classification theory of algebraic surfaces to construct the minimal model of a given smooth algebraic surface. More precisely, let X be a smooth projective surface over \mathbb and C a (−1)-curve on X (which means a smooth rational curve of self-intersection number −1), then there exists a morphism from X to another smooth projective surface Y such that the curve C has been contracted to one point P, and moreover this morphism is an isomorphism outside C (i.e., X\setminus C is isomorphic with Y\setminus P). This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve C is also called an exceptional curve of the first kind. References * *{{Citation , last1=Kollár , first1=János , last2=Mori , first2=Shigefumi , author2-link=Shigefumi Mori , author1-link=János Kollár , title=Birational geometry of algebraic varieties , publisher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
