Rational maps and function fields

For an irreducible affine variety X, its coordinate ring A is a domain, so define k(X) as its fraction field. A rational function is an element f/g with nonzero g, initially regular on the nonempty principal open D(g). Different fractions can describe the same element, and cancellation can enlarge the actual regularity domain. A fraction's displayed denominator therefore gives a valid domain, not necessarily the maximal domain. Regularity at a point means membership in the local ring, not merely a formal limit in a real picture.

An irreducible variety has intersecting nonempty opens, so rational functions can be compared on a shared dense open subset. On a reducible algebraic set, zero divisors prevent one ordinary fraction field from describing all components. One can instead study component fields or an appropriate total quotient ring, but that is a different construction. Every function-field statement in this lesson assumes irreducibility; replacing it with reducedness alone would make division by a nonzero zero divisor illegitimate.

A rational map from X to Y is represented by a regular map on a nonempty open subset of X, with two representatives equivalent if they agree on a nonempty open overlap. It is dominant when its image is dense in Y. Dominance ensures that a nonzero rational function on Y does not pull back identically to zero, so denominators remain meaningful and pullback extends to an embedding k(Y) into k(X). A map with image in a proper closed subset does not generally define this field embedding.

For irreducible varieties, an isomorphism of their function fields over k corresponds to a birational map: there are dense open subsets on which the varieties are isomorphic. One can see the dense opens by writing finitely many coordinate functions and inverse coordinate functions as fractions and excluding their denominator zero loci. This does not prove the entire varieties are isomorphic. The removed subsets can contain precisely the singularity or exceptional fibre of interest, and recovering those subsets requires local geometry that the common field does not remember.

A rational function may have an apparent singularity that disappears after applying a coordinate-ring relation. On a parabola y=x², the fraction y/x agrees with x wherever x is nonzero and therefore extends regularly across the origin. On the cusp y²=x³, the fraction y/x behaves differently: in the parameter t it equals t, whereas the coordinate ring contains only polynomials in t² and t³. There is no degree-one term available to supply a global polynomial representative. The similar-looking fractions therefore have different extension behaviour.

Even a regular map that is birational need not be finite or injective. A curve may be contracted to one point, and the image can omit other points on a closed subset. Those phenomena are invisible on a dense open where an inverse exists. To decide a stronger property, inspect the exceptional fibres and the ring extension, rather than concluding from the existence of a rational inverse. The explicit plane map below provides all three distinctions with formulas that can be verified directly.