# ag.algebraic geometry – When is a birational bijection étale?

So this question is probably not “research level”, although, for what it is worth, it is coming up in a research paper I am presently writing.

Let $$X,Y$$ be irreducible affine varieties over $$mathbb{C}$$. Suppose $$f:Xto Y$$ is a bijective birational morphism. Assume that $$X$$ is normal, but $$Y$$ is not and that both $$X,Y$$ are singular (not smooth).

Question 1: Is $$f$$ étale?

I know that $$f$$ is the normalization map, and that if $$Y$$ is normal then $$f$$ is an isomorphism in this setting. I also know that the Grothendeick classes of $$X$$ and $$Y$$ are equal (induced by $$f$$). Also, at corresponding smooth points there is a local analytic isomorphism and so at such points the map is étale.

I suspect the answer to this question is no, and that there are easy counter-examples (that I have probably seen before and am just forgetting now).

So let me ask a second question that might be more useful to me than the expected counter-example.

Question 2: Are there easy to check (think computational) conditions on $$X$$, $$Y$$, or $$f$$ that will guarantee that $$f$$ is étale?