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Let $C\rightarrow S$ be a smooth proper morphism of relative dimension 1, where $S$ is a Noetherian normal scheme. Let $D\hookrightarrow C$ be a relative effective Cartier divisor finite over $S$. Let $D_{red}$ be the corresponding reduced closed subscheme. Suppose $D_{red}$ is finite etale over $S$. Is $D_{red}$ also a (relative) effective Cartier divisor?

The idea is that given a proper curve $X/S$ and a section $g: S\rightarrow X$, we may consider a ramified Galois cover $C\rightarrow X$ over $S$ which is etale away from $g(S)$. The idea is that $D$ should be the pullback of $g(S)$ to $C$, and $D_{red}$ as a scheme should be isomorphic to the pullback of $g(S)$ to the maximal intermediate unramified cover of $X$.

Naively, I would think that it should be, though I really am not sure. I'd appreciate any related results as well.

Thank you.

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    $\begingroup$ If $S$ is normal and $D$ étale over $S$, $D$ is normal, hence reduced. $\endgroup$
    – abx
    Commented Jul 6, 2016 at 6:35
  • $\begingroup$ @abx Ah, I'm sorry I miswrote the question. I meant to say that $D_{red}$ is finite etale, not $D$ itself. I've fixed it now. $\endgroup$
    – stupid_question_bot
    Commented Jul 6, 2016 at 6:57
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    $\begingroup$ Then this is true, but it has little to do with your hypotheses. If you have an embedding $j:Y\hookrightarrow X$ with both $Y$ and $X$ smooth over a base $S$, then $j$ is a regular embedding: see EGA IV, Theorem 17.12.1. Thus if $Y$ has codimension 1 it is a Cartier divisor. $\endgroup$
    – abx
    Commented Jul 6, 2016 at 7:25
  • $\begingroup$ Ah perfect! I didn't think I'd need to use all my hypotheses, but I wanted to avoid situations where the answer is no, but the counterexample cannot appear in the particular situation I was thinking of. If you'd like to post that as an answer I'd happily accept it. Thanks! $\endgroup$
    – stupid_question_bot
    Commented Jul 6, 2016 at 17:56

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