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!

  • 1
    $\begingroup$ 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. $\endgroup$
    – KReiser
    Sep 3, 2020 at 22:39


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