Let $X$ be a normal variety over $\mathbb{C}$.

In their book *Birational geometry of algebraic varieties*, Kollár and Mori define [Definition 2.50 and 2.51] a ramified m-th cyclic cover associate to a line bundle $L$ ramified along $D \subseteq |mL|$ to be the relative spec $$Spec_X(\oplus_{i=0}^{m-1}L^{-i}).$$ Or more generally, for a rank $1$ torison free sheaf $L$, the m-th cyclic ramified cover is $$Spec_X(\oplus_{i=0}^{m-1}L^{[-i]}),$$ where $L^{[i]}$ is the double dual of $L^{\otimes{i}}$.

In the book *Singularities of the Minimal Model Program*, Kollár and Kovács give a definition of ramified cover [see Definition 2.39], which is roughly as follows:

A finite morphism of normal schemes $\pi: \tilde{X} \to X$ is called a ramified cover of degree m if there is a dense open subset $U \subseteq X$ that contains every codimension $1$ point of Sing$X$ such that the restriction $\pi_U: \tilde{U}\to U$ is etale and has degree m.

My question: Is the (general) ramified m-th cyclic cover in the sense of Kollár and Mori a special case of the ramified cover in the sense of Kollár and Kovács?

In my case, $X$ has canonical singularities, and $D$ can contain the codimension 1 point of Sing$X$. So, when choose $U$, it is inevitable to intersect $D$, hence I worry if the resulting morphism etale?

My interest in the problem is because I want to know the singularity of the ramified cyclic cover $\tilde{X}$ (in the sense of Kollár and Mori). Again, $X$ has canonical singularities, I want to know if $\tilde{X}$ has the same singularities.

By the book of Kollár and Kovács (See Page 65-65), it claims that the discrepancy does not get worse by taking a finite ramified cover (in their definition). I looked at the proof, and feel it could go through without any change for the (general ) cyclic ramified cover case. Did I miss something?

yesevery cyclic cover is a a ramified cover in the sense of Kollár and Kovács. $\endgroup$5more comments