Function field of an algebraic variety

Last updated July 20, 2024

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.

Definition for complex manifolds

In complex geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in $\mathbb {C} \cup \{\infty \}$ .) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra.

For the Riemann sphere, which is the variety $\mathbb {P} ^{1}$ over the complex numbers, the global meromorphic functions are exactly the rational functions (that is, the ratios of complex polynomial functions).

Construction in algebraic geometry

In classical algebraic geometry, we generalize the second point of view. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety V, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data as agree on the intersections of open affines. We may define the function field of V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.

Generalization to arbitrary scheme

In the most general setting, that of modern scheme theory, we take the latter point of view above as a point of departure. Namely, if $X$ is an integral scheme, then for every open affine subset $U$ of $X$ the ring of sections ${\mathcal {O}}_{X}(U)$ on $U$ is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the stalk of the generic point of $X$ . Thus the function field of $X$ is just the stalk of its generic point. This point of view is developed further in function field (scheme theory). See RobinHartshorne ( 1977 ).

Geometry of the function field

If V is a variety defined over a field K, then the function field K(V) is a finitely generated field extension of the ground field K; its transcendence degree is equal to the dimension of the variety. All extensions of K that are finitely generated as fields over K arise in this way from some algebraic variety. These field extensions are also known as algebraic function fields over K.

Properties of the variety V that depend only on the function field are studied in birational geometry.

Examples

The function field of a point over K is K.

The function field of the affine line over K is isomorphic to the field K(t) of rational functions in one variable. This is also the function field of the projective line.

Consider the affine algebraic plane curve defined by the equation $y^{2}=x^{5}+1$ . Its function field is the field K(x,y), generated by elements x and y that are transcendental over K and satisfy the algebraic relation $y^{2}=x^{5}+1$ .

Related Research Articles

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.

In commutative algebra, the prime spectrum of a commutative ring R is the set of all prime ideals of R, and is usually denoted by $; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .$

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space.

<span class="mw-page-title-main">Projective space</span> Completion of the usual space with "points at infinity"

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem 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.

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.

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation $h (x, y, t) = 0$ can be restricted to the affine algebraic plane curve of equation $h (x, y, 1) = 0$ . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In mathematics, a transcendental extension $is a field extension such that there exists an element in the field that is transcendental over the field; that is, an element that is not a root of any univariate polynomial with coefficients in . In other words, a transcendental extension is a field extension that is not algebraic. For example, and are both transcendental extensions of$

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field $k$ of some family of polynomials in the polynomial ring $An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.$

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

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. The coefficients of the polynomials need not be rational numbers; they may be taken in any field $K$ . In this case, one speaks of a rational function and a rational fraction over $K$ . The values of the variables may be taken in any field $L$ containing $K$ . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is $L$ .

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

The sheaf of rational functionsK_X of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties, such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, K_X(U) is the set of fractions of regular functions on U. Despite its name, K_X does not always give a field for a general scheme X.

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension $n - 1$ , which is embedded in an ambient space of dimension $n$ , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally, and sometimes globally.

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. Both are derived from the notion of divisibility in the integers and algebraic number fields.

In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism.

In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.

In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials.

This is a glossary of algebraic geometry.

In algebraic geometry, a new scheme can be obtained by gluing existing schemes through gluing maps.

References

David M. Goldschmidt (2002). Algebraic Functions and Projective Curves. Graduate Texts in Mathematics. Vol. 215. Springer-Verlag. ISBN 0-387-95432-5.
Hartshorne, Robin (1977), Algebraic Geometry , Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052 , section II.3 First Properties of Schemes exercise 3.6

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.