# Function field of an algebraic variety

In algebraic geometry, 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.

## Definition for complex manifolds

In complex algebraic 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 **C**u{∞}.) 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 **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 which 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 every open affine subset *U* of *X* the ring of sections 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 local ring of the generic point of *X*. Thus the function field of *X* is just the local ring of its generic point. This point of view is developed further in function field (scheme theory). See Robin Hartshorne (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 plane curve defined by the equation . 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 .

## See also

- Function field (scheme theory): a generalization
- Algebraic function field
- Cartier divisor

## References

- David M. Goldschmidt (2002).
*Algebraic Functions and Projective Curves*. Graduate Texts in Mathematics.**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