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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] |
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Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Smooth Morphism
In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) means that each geometric fiber of ''f'' is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If ''S'' is the spectrum of an algebraically closed field and ''f'' is of finite type, then one recovers the definition of a nonsingular variety. A singular variety is called smoothable if it can be put in a flat family so that the nearby fibers are all smooth. Such a family is called a smoothning of the variety. Equivalent definitions There are many equivalent definitions of a smooth morphism. Let f: X \to S be locally of finite presentation. Then the following are equivalent. # ''f'' is smooth. # ''f'' is formally smooth (see below). # ''f'' is flat and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Euler Sequence
In mathematics, the Euler sequence is a particular exact sequence of sheaves on ''n''-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an (n+1)-fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Statement Let \mathbb^n_A be the ''n''-dimensional projective space over a commutative ring ''A''. Let \Omega^1 = \Omega^1_ be the sheaf of 1-differentials on this space, and so on. The Euler sequence is the following exact sequence of sheaves on \mathbb^n_A: 0 \longrightarrow \Omega^1 \longrightarrow \mathcal(-1)^ \longrightarrow \mathcal \longrightarrow 0. The sequence can be constructed by defining a homomorphism S(-1)^ \to S, e_i \mapsto x_i with S = A _0, \ldots, x_n/math> and e_i = 1 in degree 1, surjective in degrees \geq 1, and checking that locally on the n+1 standard c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Chow Ring
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. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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First Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sections a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Veronese Surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese (1854–1917). Its generalization to higher dimension is known as the Veronese variety. The surface admits an embedding in the four-dimensional projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective space is called a Steiner surface. Definition The Veronese surface is the image of the mapping :\nu:\mathbb^2\to \mathbb^5 given by :\nu: :y:z\mapsto ^2:y^2:z^2:yz:xz:xy/math> where :\cdots/math> denotes homogeneous coordinates. The map \nu is known as the Veronese embedding. Motivation The Veronese surface arises naturally in the study of conics. A conic is a degree 2 plane curve, thus defined by an equation: :Ax^2 + Bxy + Cy^2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Scheme-theoretic Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship between given quantities. The plural of ''formula'' can be either ''formulas'' (from the most common English plural noun form) or, under the influence of scientific Latin, ''formulae'' (from the original Latin). In mathematics In mathematics, a formula generally refers to an equation or inequality relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Projection Formula
In algebraic geometry, the projection formula states the following: For a morphism f:X\to Y of ringed spaces, an \mathcal_X-module \mathcal and a locally free \mathcal_Y-module \mathcal of finite rank, the natural maps of sheaves :R^i f_* \mathcal \otimes \mathcal \to R^i f_* (\mathcal \otimes f^* \mathcal) are isomorphisms. There is yet another projection formula in the setting of étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu .... See also * References Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |