Birational geometry, the Glossary
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.[1]
Table of Contents
58 relations: Abundance conjecture, Algebraic curve, Algebraic geometry, Algebraic variety, Ample line bundle, Annals of Mathematics, Betti number, Blowing up, Cambridge University Press, Canonical bundle, Canonical singularity, Caucher Birkar, Chow's lemma, Cotangent bundle, Cremona group, Del Pezzo surface, Enriques–Kodaira classification, Exterior algebra, Fano variety, Flip (mathematics), Function field of an algebraic variety, Fundamental group, Guido Castelnuovo, Heisuke Hironaka, Hodge theory, Irreducible component, Isomorphism, Italian school of algebraic geometry, János Kollár, Journal of the American Mathematical Society, K-stability of Fano varieties, Kähler–Einstein metric, Kodaira dimension, Line bundle, Map (mathematics), Matematicheskii Sbornik, Mathematics, Max Noether, Minimal model program, Moduli space, Nef line bundle, Nicholas Shepherd-Barron, Number line, Polynomial, Projective space, Projective variety, Pythagorean triple, Quadric (algebraic geometry), Rational function, Rational mapping, ... Expand index (8 more) »
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.
See Birational geometry and Abundance conjecture
Algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables.
See Birational geometry and Algebraic curve
Algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.
See Birational geometry and Algebraic geometry
Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.
See Birational geometry and Algebraic variety
Ample line bundle
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two).
See Birational geometry and Ample line bundle
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Birational geometry and Annals of Mathematics
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
See Birational geometry and Betti number
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.
See Birational geometry and Blowing up
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge.
See Birational geometry and Cambridge University Press
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.
See Birational geometry and Canonical bundle
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.
See Birational geometry and Canonical singularity
Caucher Birkar
Caucher Birkar (translit; born Fereydoun Derakhshani (فریدون درخشانی); July 1978) is an Iranian Kurd mathematician and a professor at Tsinghua University.
See Birational geometry and Caucher Birkar
Chow's lemma
Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry.
See Birational geometry and Chow's lemma
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.
See Birational geometry and Cotangent bundle
Cremona group
In algebraic geometry, the Cremona group, introduced by, is the group of birational automorphisms of the n-dimensional projective space over a field It is denoted by Cr(\mathbb^n(k)) or Bir(\mathbb^n(k)) or Cr_n(k).
See Birational geometry and Cremona group
Del Pezzo surface
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class.
See Birational geometry and Del Pezzo surface
Enriques–Kodaira classification
In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space.
See Birational geometry and Enriques–Kodaira classification
Exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v.
See Birational geometry and Exterior algebra
Fano variety
In algebraic geometry, a Fano variety, introduced by Gino Fano in, is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient projective space.
See Birational geometry and Fano variety
Flip (mathematics)
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring.
See Birational geometry and Flip (mathematics)
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
See Birational geometry and Function field of an algebraic variety
Fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space.
See Birational geometry and Fundamental group
Guido Castelnuovo
Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician.
See Birational geometry and Guido Castelnuovo
Heisuke Hironaka
is a Japanese mathematician who was awarded the Fields Medal in 1970 for his contributions to algebraic geometry.
See Birational geometry and Heisuke Hironaka
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.
See Birational geometry and Hodge theory
Irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets.
See Birational geometry and Irreducible component
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.
See Birational geometry and Isomorphism
Italian school of algebraic geometry
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935.
See Birational geometry and Italian school of algebraic geometry
János Kollár
János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry.
See Birational geometry and János Kollár
Journal of the American Mathematical Society
The Journal of the American Mathematical Society (JAMS), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society.
See Birational geometry and Journal of the American Mathematical Society
K-stability of Fano varieties
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds.
See Birational geometry and K-stability of Fano varieties
Kähler–Einstein metric
In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric.
See Birational geometry and Kähler–Einstein metric
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X.
See Birational geometry and Kodaira dimension
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space.
See Birational geometry and Line bundle
Map (mathematics)
In mathematics, a map or mapping is a function in its general sense.
See Birational geometry and Map (mathematics)
Matematicheskii Sbornik
Matematicheskii Sbornik (Математический сборник, abbreviated Mat. Sb.) is a peer reviewed Russian mathematical journal founded by the Moscow Mathematical Society in 1866.
See Birational geometry and Matematicheskii Sbornik
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Birational geometry and Mathematics
Max Noether
Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions.
See Birational geometry and Max Noether
Minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties.
See Birational geometry and Minimal model program
Moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
See Birational geometry and Moduli space
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.
See Birational geometry and Nef line bundle
Nicholas Shepherd-Barron
Nicholas Ian Shepherd-Barron, FRS (born 17 March 1955), is a British mathematician working in algebraic geometry.
See Birational geometry and Nicholas Shepherd-Barron
Number line
In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.
See Birational geometry and Number line
Polynomial
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.
See Birational geometry and Polynomial
Projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity.
See Birational geometry and Projective space
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.
See Birational geometry and Projective variety
Pythagorean triple
A Pythagorean triple consists of three positive integers,, and, such that.
See Birational geometry and Pythagorean triple
Quadric (algebraic geometry)
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field.
See Birational geometry and Quadric (algebraic geometry)
Rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials.
See Birational geometry and Rational function
Rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties.
See Birational geometry and Rational mapping
Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
See Birational geometry and Rational number
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set \ of indeterminates, where d is the dimension of the variety.
See Birational geometry and Rational variety
Resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, which is a non-singular variety W with a proper birational map W→V.
See Birational geometry and Resolution of singularities
Segre embedding
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety.
See Birational geometry and Segre embedding
Singular point of an algebraic variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined.
See Birational geometry and Singular point of an algebraic variety
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.
See Birational geometry and Springer Science+Business Media
Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point.
See Birational geometry and Stereographic projection
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties.
See Birational geometry and Zariski topology
References
[1] https://en.wikipedia.org/wiki/Birational_geometry
Also known as Birational, Birational automorphism, Birational classification, Birational equivalence, Birational map, Birational mapping, Birational morphism, Birational transformation, Birationally equivalent.
, Rational number, Rational variety, Resolution of singularities, Segre embedding, Singular point of an algebraic variety, Springer Science+Business Media, Stereographic projection, Zariski topology.