# ramification locus for finite morphism and Abhyankar's Lemma

I want to ask given a finite morphism between projective varieties $$f:X\rightarrow Y$$. What is exactly the ramification locus $$\Delta(X/Y)$$. If $$X$$, $$Y$$, $$f$$ are smooth, then I can more or less understand it.

Q1: There should be strict definition of $$\Delta(X/Y)$$ (Any reference?). Also, is there a good example to see $$\Delta(X/Y)$$ with $$X,Y$$ singular variety.

Q2: I saw a theorem, which says that if $$f:X\rightarrow Y$$ is finite morphism,$$X$$ normal, $$Y$$ smooth and $$\Delta(X/Y)$$ simple normal crossing, then $$X$$ has quotient singularity. The proof is refered to Abhyankar's Lemma (in Germany), I wonder how the Abhyankar's Lemma can be applied here. Can I get a rough explanation! (Reference in English is also helpful) Thanks!

• The ramification locus is just the support of the sheaf $\Omega_{X/Y}$ on $X$. This is in many texts. To get examples with $X,Y$ singular, all you need to do is to compute what $\Omega_{X/Y}$ for a map of singular varieties $X\to Y$ and see where it's supported. Sep 3, 2020 at 22:39