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The cusp normalization is bijective but not an isomorphism

Statement refuted

False claim: a finite birational morphism of classical varieties that is bijective is an isomorphism.

Facts & Assumptions

Assume the Axiom of Choice.

Given: AC, an algebraically closed field $k$ of characteristic not two, the cusp $X = V (y^{2} - x^{3}) \subseteq A^{2}$ , and its normalization $ν ⁣ : A^{1} \to X$ , $t \mapsto (t^{2}, t^{3})$ .

[F1]

The normalization $ν$ is finite and birational, and the cusp is its unique non-normal point; its pullback on coordinate rings is the inclusion $k [t^{2}, t^{3}] ↪ k [t]$ , which is injective but not surjective, so $ν$ is not an isomorphism (Normalizing the cuspidal plane curve, Finite morphisms of classical varieties, Irreducible affine varieties are birational exactly when their function fields are isomorphic).

[F2]

The same computation records that $ν$ is a bijection on points: it is the parametrization $t \mapsto (t^{2}, t^{3})$ , and every point of the cusp is the image of a unique parameter (Normalizing the cuspidal plane curve).

[F3]

A finite birational morphism onto a normal target is an isomorphism; normality of the target is a hypothesis, and the cusp is a non-normal target (A finite birational morphism onto a normal variety is an isomorphism, Normal points and normal varieties).

[F7]

AC is inherited through the classical localization, normalization, or finite-morphism suppliers cited above (The Axiom of Choice).

Counterexample

1.1F1F2givenF7

By [F1] the normalization $ν$ is finite and birational, and by [F2] it is bijective but not an isomorphism: its pullback $k [t^{2}, t^{3}] ↪ k [t]$ misses $t$ , so this pullback is not an isomorphism.

1.2F1F3given

The target of $ν$ is the cusp, which is not normal at its singular point by [F1]; the isomorphism criterion [F3] therefore does not apply, exactly because its target-normality hypothesis fails.

2.1F1F2F3step 1.1step 1.2∎

Hence finite plus birational plus bijective does not imply isomorphism: the cusp normalization is a counterexample, and bijectivity cannot replace the normality hypothesis in the finite-birational-to-normal isomorphism lemma. The failure is isolated precisely at the non-normal point of the target.

Depends on

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Sources

J. S. Milne, Algebraic Geometry (2025 version), Ch. 8 Example 8.6(a): the cusp parametrization is bijective but not an isomorphism (standard reference, not scraped)