rational map

A rational map is a morphism that is allowed to misbehave at a few bad spots. Like the function 1/x, which is perfectly good everywhere except at the origin, a rational map is given by ratios of polynomials and is defined only on the open set where the denominators do not vanish. We tolerate this because such maps capture the birational geometry that classifies varieties up to a 'small surgery'.

Formally, on irreducible varieties a rational map from X to Y is an equivalence class of pairs (U, f) where U is a nonempty (hence dense) open subset of X and f is a morphism U to Y, two pairs being identified when they agree on the overlap. Because X is irreducible, this glues to a maximal domain of definition, an open set whose complement is the locus of indeterminacy. One writes a dashed arrow X to Y to signal that the map may be undefined somewhere.

Rational maps need not be composable, since the image of one may land entirely in the indeterminacy of the next; but dominant rational maps (those with dense image) do compose. A dominant rational map X to Y induces a field homomorphism between function fields k(Y) to k(X), and this is an equivalence: dominant rational maps correspond to k-embeddings of function fields. A birational map — a rational map with a rational inverse — corresponds to an isomorphism of function fields, the central equivalence relation of birational geometry.