3.5 Resolving rational maps

We will introduce a systematic procedure for replacing rational maps by morphisms
defined on a smaller variety. This will be used to compute the image of a rational map
in Chapter 4:
Proposition 3.47 Let V and W be affine varieties realized in An(k) and Am(k)
respectively. Consider a rational map ρ : V W obtained from a map
An(k) An(k)
(x1, . . . , xn) → ( f1/g1, . . . , fm/gm)
admissible on V. Write g = g1 . . . gm so that ρ is well-defined over the open set
U = {v ∈ V : g(v) = 0}. There is an affine variety Vg and morphisms π : Vg → V
and φ : Vg → W, with the following properties:

1. π(Vg) = U;
2. there is a rational map ψ : V Vg, well-defined on U, such that π ◦ ψ = IdU and
ψ ◦ π = IdVg ;
Vg
f
W
V
p r
3. φ = ρ ◦ π.
Thus π is a birational morphism, i.e., a morphism which admits an inverse rational
map.