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Abundance Conjecture
In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonical bundle K_X is nef, then K_X is semi-ample. Important cases of the abundance conjecture have been proven by Caucher Birkar Caucher Birkar ( ku, کۆچەر بیرکار, lit=migrant mathematician, translit=Koçer Bîrkar; born Fereydoun Derakhshani ( fa, فریدون درخشانی); July 1978) is an Iranian peoples, Iranian Kurds, Kurdish mathematician and a profes .... References * * Algebraic geometry Birational geometry Unsolved problems in 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 from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kawamata Log Terminal 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] |
Canonical 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 is the determinant bundle of holomorphic ''n''-forms on ''V''. This is the dualising object for Serre duality on ''V''. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor ''K'' on ''V'' giving rise to the canonical bundle — it is an equivalence class for linear equivalence on ''V'', and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −''K'' with ''K'' canonical. The anticanonical bundle is the corresponding inverse bundle ω−1. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that ''X'' is a smooth variety and that ''D'' is a smooth divisor on ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nef Line Bundle
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. Definition More generally, a line bundle ''L'' on a proper scheme ''X'' over a field ''k'' is said to be nef if it has nonnegative degree on every (closed irreducible) curve in ''X''. (The degree of a line bundle ''L'' on a proper curve ''C'' over ''k'' is the degree of the divisor (''s'') of any nonzero rational section ''s'' of ''L''.) A line bundle may also be called an invertible sheaf. The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" and "numerically effective", as well as for the phrase "num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caucher Birkar
Caucher Birkar ( ku, کۆچەر بیرکار, lit=migrant mathematician, translit=Koçer Bîrkar; born Fereydoun Derakhshani ( fa, فریدون درخشانی); July 1978) is an Iranian Kurdish mathematician and a professor at Tsinghua University and University of Cambridge. Birkar is an important contributor to modern birational geometry. In 2010 he received the Leverhulme Prize in mathematics and statistics for his contributions to algebraic geometry, and in 2016, shared the AMS Moore Prize for the article "Existence of minimal models for varieties of log general type". He was awarded the Fields Medal in 2018, "for his proof of boundedness of Fano varieties and contributions to the minimal model program". In his office at the University, Birkar has two photographs of Alexander Grothendieck, his favorite mathematician, who like Birkar, was a refugee and Fields medalist. Early life and education Birkar is a Kurd, born in 1978 in Marivan County, Kurdistan province, Iran, on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...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 of the ... [...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 from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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