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?
Best Answer
Ok, so based on the discussion in the comments, maybe I should put this into an answer. I think the confusion comes from the phrase
every codimension 1 point of $\text{Sing }X$.
What the authors Kollár and Kovács mean here is to consider
every point of $\text{Sing }X$ that is also a codimension $1$ point of $X$.
I agree that this could be interpreted in other ways, but this is what the authors mean (I'm sure Sándor will agree if he sees this).
Given this, and that $X$ is normal over $\mathbb{C}$ (or any field of characteristic zero), it is easy to see that every cyclic cover is a ramified cover in the sense of Kollár-Kovács (as I think you already see). As pointed out in the comments, if $X$ is non-normal, or if we are working in characteristic $p > 0$, life becomes more complicated.
Your particular situation
You had $X$ with canonical singularities and $\widetilde{X}$ a ramified cyclic cover. Then it is easy to see that $(X, -\text{(Ramification Divisor)})$ also has canonical singularities (notice we have a non-effective divisor here). For a proof simply see Kollár-Mori 5.20(3).