Birational geometry (nonfiction)

The circle is birationally equivalent to the line. One birational map between them is a stereographic projection onto the z=0 plane, pictured here.

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

A rational map from one variety (understood to be an irreducible component) X to another variety Y, written as a dashed arrow X ⇢ Y, is defined as a morphism from a nonempty open subset U of X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always the complement of a lower-dimensional subset of X. Concretely, a rational map can be written in coordinates using rational functions.

A birational map from X to Y is a rational map f: X ⇢ Y such that there is a rational map Y ⇢ X inverse to f. A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y. In this case, X and Y are said to be birational, or birationally equivalent. In algebraic terms, two varieties over a field k are birational if and only if their function fields are isomorphic as extension fields of k.

A special case is a birational morphism f: X → Y, meaning a morphism which is birational. That is, f is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of X to points in Y.

A variety X is said to be rational if it is birational to affine space (or equivalently, to projective space) of some dimension. Rationality is a very natural property: it means that X minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.

Fiction cross-reference

• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematician
• Mathematics

Nonfiction cross-reference

• Abundance conjecture (nonfiction) - 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 {\displaystyle X} X with Kawamata log terminal singularities over a field {\displaystyle k} k if the canonical bundle {\displaystyle K_{X}} K_{X} is nef, then {\displaystyle K_{X}} K_{X} is semi-ample.
• Algebraic geometry (nonfiction) - 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.
• Algebraic variety (nonfiction) - the central objects of study in algebraic geometry. 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.
• Function field of an algebraic variety (nonfiction) - the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic 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.
• Irreducible component (nonfiction) - in algebraic geometry, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation XY = 0 is the union of the two lines X = 0 and Y = 0. Thus an algebraic set is irreducible if it is not the union of two proper algebraic subsets. It is a fundamental theorem of classical algebraic geometry that every algebraic set is the union of a finite number of irreducible algebraic subsets (varieties), and that, if one removes those subsets contained in another one, this decomposition is unique. The elements of this unique decomposition are called irreducible components.
• Mathematics (nonfiction)
• Polynomial (nonfiction) - an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
• Projective space (nonfiction) - the concept of a projective space originated from the visual effect of perspective, where parallel lines seems to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore other definitions are generally preferred. There are two classes of definitions:
  • In synthetic geometry, point and lines are primitive entities the are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to at the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.
  • Using linear algebra, a projective space of dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently, it is the quotient set of V \ {0} by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of V in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
• Rational function (nonfiction) - any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
• Rationality question (nonfiction) - asks whether a given field extension is rational, in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described as purely transcendental.
• Rational mapping (nonfiction) - in algebraic geometry, a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
• Rational variety (nonfiction) - an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K.
• Stereographic projection (nonfiction) - In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.
• Zariski topology (nonfiction) - in algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.

External links:

• Birational geometry @ Wikipedia