Regular function

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 (mathematics) K over which V is defined.

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.

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.

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.

In fact taking the 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.

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.