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Mori 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] |
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] |
Contraction 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 , if it exists, is a contract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Rational Surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birational geometry, birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces, and were the first surfaces to be investigated. Structure Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces Σ''r'' for ''r'' = 0 or ''r'' ≥ 2. Invariants: The plurigenera are all 0 and the fundamental group is trivial. Homological mirror symmetry#Hodge diamond, Hodge diamond: where ''n'' is 0 for the projective plane, and 1 for Hirzebruch surfaces and greater than 1 for other rational surfaces. The Picard group is the odd unimodular lattice I1,''n'', except for the Hirzebruch surfaces Σ2''m'' when it is the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (; born Fereydoun Derakhshani (، ); July 1978) is a UK-based Iranian Kurdish mathematician (born in Iran) and a professor at Tsinghua University. Birkar is an important contributor to modern birational geometry. In 2010 he re .... References * * Algebraic geometry Birational geometry Unsolved problems in geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James McKernan
James McKernan (born 1964) is a mathematician, and a professor of mathematics at the University of California, San Diego. He was a professor at MIT from 2007 until 2013. Education McKernan was educated at The Campion School and Trinity College, Cambridge, before going on to earn his Ph.D. from Harvard University in 1991. His dissertation, ''On the Hyperplane Sections of a Variety in Projective Space'', was supervised by Joe Harris. Recognition McKernan was the joint winner of the Cole Prize in 2009, and joint recipient of the Clay Research Award in 2007. Both honors were received jointly with his colleague Christopher Hacon. He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Algebraic Geometry". He was the joint winner (with Christopher Hacon) of the 2018 Breakthrough Prize in Mathematics. He was elected as a Fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of profess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Hacon
Christopher Derek Hacon (born 14 February 1970) is a mathematician with British, Italian and US nationalities. He is currently distinguished professor of mathematics at the University of Utah where he holds a Presidential Endowed Chair. His research interests include algebraic geometry. Hacon was born in Manchester, but grew up in Italy where he studied at the Scuola Normale Superiore and received a degree in mathematics at the University of Pisa in 1992. He received his doctorate from the University of California, Los Angeles in 1998, under supervision of Robert Lazarsfeld. Awards and honors In 2007, he was awarded a Clay Research Award for his work, joint with James McKernan, on "the birational geometry of algebraic varieties in dimension greater than three, in particular, for ninductive proof of the existence of flips." In 2009, he was awarded the Cole Prize for outstanding contribution to algebra, along with McKernan. He was an invited speaker at the International Cong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caucher Birkar
Caucher Birkar (; born Fereydoun Derakhshani (، ); July 1978) is a UK-based Iranian Kurdish mathematician (born in Iran) and a professor at Tsinghua University. 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. Birkar maintains strong ties to his Kurdish heritage and actively encourages Kurdish identity while also separating it from nationalism and politics. According to Birkar, his strong Kurdish identity is not a part of na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vyacheslav Shokurov
Vyacheslav Vladimirovich Shokurov (; born 18 May 1950) is a Russian mathematician best known for his research in algebraic geometry. The proof of the Noether–Enriques–Petri theorem, the cone theorem, the existence of a line on smooth Fano varieties and, finally, the existence of log flips—these are several of Shokurov's contributions to the subject. Early years In 1968 Shokurov became a student at the Faculty of Mechanics and Mathematics of Moscow State University. Already as an undergraduate, Shokurov showed himself to be a mathematician of outstanding talent. In 1970, he proved the scheme analog of the Noether–Enriques–Petri theorem, which later allowed him to solve a Schottky-type problem for the polarized Prym varieties, and to prove the existence of a line on smooth Fano varieties. Upon his graduation Shokurov entered the Ph.D. program in Moscow State University under the supervision of Yuri Manin. At this time Shokurov studied the geometry of Kuga varietie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flip (algebraic Geometry)
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the poin ... along a relative canonical model, relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions. The minimal model program The minimal model program can be summarised very briefly as follows: given a variety X, we construct a sequence of contracting morphism, contractions X = X_1\rightarrow X_2 \rightarrow \cdots \rightarrow X_n , each of which contracts some curves on which the canonical divisor K_ is negative. Eventually, K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shigefumi Mori
is a Japanese mathematician, known for his work in algebraic geometry, particularly in relation to the classification of three-folds. He won the Fields Medal in 1990. Career Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian Varieties" under Masayoshi Nagata at Kyoto University in 1978. He was a visiting professor at Harvard University during 1977–1980, the Institute for Advanced Study in 1981–82, Columbia University 1985–87 and the University of Utah for periods during 1987–89 and again during 1991–92. He has been a professor at Kyoto University since 1990. Work He generalized the classical approach to the classification of algebraic surfaces to the classification of algebraic three-folds. The classical approach used the concept of minimal model (birational geometry), minimal models of algebraic surfaces. He found that the concept of minimal model (birational geometry), minimal models can be applied to three-folds as well if we allow some Singular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone Of Curves
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, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartier Divisor
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] |
