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1.1. Affine varieties 13 b) For an affine variety W ⊂ An C , a rational map ϕ : V → W is an n-tuple (ϕ1, . . . , ϕn) of rational functions ϕ1,...,ϕn ∈ C(V ) such that ϕ(P ) := (ϕ1(P ), . . . , ϕn(P )) ∈ W for all P ∈ dom(ϕ). A rational map ϕ : V → W induces a C-algebra morphism ϕ∗ : C[W ] → C(V ) given by g → g ◦ ϕ. To determine when it is possible to extend this morphism to C(W ), in the case when W is also irreducible, we make the following definition. Definition 1.1.22. A rational map ϕ : V → W is called dominant if ϕ(dom(ϕ)) is a Zariski dense subset of W . Proposition 1.1.23. For irreducible affine varieties V and W , the following hold. a) Every dominant rational map ϕ : V → W induces a C-linear morphism ϕ∗ : C(W ) → C(V ). b) If f : C(W ) → C(V ) is a field homomorphism which is C-linear, then there exists a unique dominant rational map ϕ : V → W with f = ϕ∗. c) If ϕ : V → W and ψ : W → X are dominant, then ψ ◦ ϕ : V → X is also dominant and (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ∗. Proof. a) If ϕ is defined by ϕ1,...,ϕn, with ϕi ∈ C(V ), then for g ∈ C[W ], g(ϕ1,...,ϕn) ∈ C(V ). Hence ϕ induces a C-algebra morphism ϕ∗ : C[W ] → C(V ). Now if ϕ∗(g) = 0, then g vanishes on ϕ(dom(ϕ)), which is dense in W , so g = 0, hence ϕ∗ is injective, so it extends to C(W ). b) If W ⊂ An, then the restrictions x1,x2,...,xn of the coordinate functions to W , generate C(W ). Let ϕi := f(xi) ∈ C(V ) and ϕ := (ϕ1, . . . , ϕn). Then ϕ defines a rational map from V to W and f = ϕ∗ by construction. Now ϕ∗ |C[W ] = f|C[W ] is injective, so ϕ is dominant, as otherwise there will be nonzero elements in C[W ] vanishing at ϕ(dom(ϕ)), hence in Ker ϕ∗. c) As ϕ(dom(ϕ)) ∩ dom ψ = ∅, the composition makes sense and both state- ments are clear. Definition 1.1.24. Let V, W be irreducible affine varieties. A rational map ϕ : V → W is called birational (or a birational equivalence) if there is a rational map ψ : W → V with ϕ ◦ ψ = Iddom(ψ) and ψ ◦ ϕ = Iddom(ϕ). Definition 1.1.25. Two irreducible varieties V and W are said to be bira- tionally equivalent if there is a birational equivalence ϕ : V → W . Proposition 1.1.26. Let V, W be irreducible affine varieties. For a rational map ϕ : V → W , the following statements are equivalent. a) ϕ is birational. b) ϕ is dominant and ϕ∗ : C(W ) → C(V ) is an isomorphism.