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Karl Schwede
Karl Schwede
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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).

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.

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).
Source Link
Karl Schwede
Karl Schwede
  • 20.5k
  • 3
  • 53
  • 98

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.