In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined. Mathematics is the study of quantity, structure, space and change. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

For example, if V is the affine line over K, the regular functions on V make up a commutative ring, under pointwise multiplication of functions, isomorphic with the polynomial ring in one variable over K. In other words, the regular functions are just polynomials in some natural parameter on the affine line. In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...

More generally, for any affine variety V, the regular functions make up the coordinate ring of V, often written K[V]. This can be expressed in other ways. A regular function is the same as a morphism to the affine line, or in the language of scheme theory a global section of the structure sheaf. This article is about algebraic varieties. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

The reason for looking at regular functions becomes more apparent when one allows V to be a projective variety. Then regular functions on V become rare. For example morphisms from a projective space to the affine line must be constant: regular functions on a projective space are constant functions. The same is true for any connected projective variety. This article is about algebraic varieties. ... In mathematics, a projective space is a fundamental construction from any vector space. ...

In fact taking the function field K(V) of an irreducible algebraic curve V, the functions F in the function field may all be realised as morphisms from V to the projective line over K. The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective varieties). That is, unless F is actually constant, we have to attribute to F the value ∞ at some points of V. Now in some sense F is no worse behaved at those points than anywhere else: ∞ is just the chosen point at infinity on the projective line, and by using a Möbius transformation we can move it anywhere we wish. But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants. In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the projective line is a fundamental example of an algebraic curve. ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius. ...

For those reasons, the larger class of rational functions are constantly used in algebraic geometry. For the needs of birational geometry, more generally, morphisms are replaced with morphisms defined on open dense subsets. This brings fresh phenomena in dimension ≥ 1. In mathematics, a rational function is a ratio of polynomials. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...