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Multiplier Ideal
In commutative algebra, the multiplier ideal associated to a sheaf of ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ... over a complex variety and a real number ''c'' consists (locally) of the functions ''h'' such that : \frac is locally integrable, where the ''f''''i'' are a finite set of local generators of the ideal. Multiplier ideals were independently introduced by (who worked with sheaves over complex manifolds rather than ideals) and , who called them adjoint ideals. Multiplier ideals are discussed in the survey articles , , and . Algebraic geometry In algebraic geometry, the multiplier ideal of an effective \mathbb- divisor measures singularities coming from the fractional parts of ''D''. Multiplier ideals are often applied in tandem with vanishing theorems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Integrable Function
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions. Definition Standard definition .See for example and . Let be an open set in the Euclidean space \mathbb^n and be a Lebesgue measurable function. If on is such that : \int_K , f , \, \mathrmx <+\infty, i.e. its |
Divisor (algebraic Geometry)
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Vanishing Theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implications for the group with index ''q'' = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem. The complex analytic case The statement of Kunihiko Kodaira's result is that if ''M'' is a compact Kähler manifold of complex dimension ''n'', ''L'' any holomorphic line bundle on ''M'' that is positive, and ''KM'' is the canonical line bundle, then ::: H^q(M, K_M\otimes L) = 0 for ''q'' > 0. Here K_M\otimes L stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of H^q(M, L^) for ''q'' ''n''. The algebraic case The Kodaira vanishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kawamata–Viehweg Vanishing Theorem
In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982. The theorem states that if ''L'' is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ... ''K'', then the coherent cohomology groups ''H''''i''(''L''⊗''K'') vanish for all positive ''i''. References * * Theorems in algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log Resolution
Log most often refers to: * Trunk (botany), the stem and main wooden axis of a tree, called logs when cut ** Logging, cutting down trees for logs ** Firewood, logs used for fuel ** Lumber or timber, converted from wood logs * Logarithm, in mathematics Log, LOG or LoG may also refer to: Arts, entertainment and media * ''Log'' (magazine), an architectural magazine * ''The Log'', a boating and fishing newspaper published by the Duncan McIntosh Company * Lamb of God (band) or LoG, an American metal band * The Log, an electric guitar by Les Paul * Log, a fictional product in '' The Ren & Stimpy Show'' * The League of Gentlemen or LoG, a British comedy show. Places * Log, Russia, the name of several places * Log, Slovenia, the name of several places Science and mathematics * Logarithm, a mathematical function * Log file, a computer file in which events are recorded * Laplacian of Gaussian or LoG, an algorithm used in digital image processing Other uses * Logbook, or log, a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Singularity
In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities. Definition Suppose that ''Y'' is a normal variety such that its canonical class ''K''''Y'' is Q-Cartier, and let ''f'':''X''→''Y'' be a resolution of the singularities of ''Y''. Then :\displaystyle K_X = f^*(K_Y)+\sum_i a_iE_i where the sum is over the irreducible exceptional divisors, and the ''a''''i'' are rational numbers, called the discrepancies. Then the singularities of ''Y'' are called: :terminal if ''a''''i'' > 0 for all ''i'' :canonical if ''a''''i'' ≥ 0 for all ''i'' :log terminal if ''a''''i'' > −1 for all ''i'' :log canonical if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Test Ideal
A test ideal is a positive characteristic analog of a multiplier ideal In commutative algebra, the multiplier ideal associated to a sheaf of ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with ... in, say, the field of complex numbers. Test ideals are used in the study of singularities in algebraic geometry in positive characteristic. References Commutative algebra {{commutative-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
