Let $D$ and $S$ be two regular schemes and let $D$ be a divisor of $S$. Let $C \to S$ be a finite flat morphism, branched along $D$. Is $C$ regular as well?
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1$\begingroup$ I am not sure what branched means. But if $D$ is an affine line over $\spec \mathbb C$, $S$ a closed point of $D$ (so $D$ and $S$ are clearly regular), and $C$ the union of two copies of $D$, intersecting at the point $S$, then the natural morphism $C \rightarrow D$ (the one sending each irreducible components of $D$ to $C$ by the identity map) is finite flat, and étale outside $S$, while $C$ is not regular. $\endgroup$ – Joël May 8 '13 at 13:10

2$\begingroup$ Joel: right. Anyway, in your example the schematic branch locus is not reduced (is one point counted with multiplicity two). Indeed, your cover is the limit of a family of flat double covers $f_t \colon \mathbb{P}^1 \to \mathbb{P}^1$, all branched in two points, when the two points come together. $\endgroup$ – Francesco Polizzi May 8 '13 at 13:41

1$\begingroup$ Ah okay. So what is the definition of the schematic branch locus? $\endgroup$ – Joël May 8 '13 at 14:41

$\begingroup$ @Francesco: We should be able to see that the schematic branch locus has multiplicity two by looking at the ring map $\mathbb{C}[x]\rightarrow\mathbb{C}[x,y]/(x^2y^2)$. So Joël's question is pertinent: what is the definition of branch locus? If this is not a counterexample for the OP's definition of branched along $D$ (because it is actually branched along $2D$, which is not regular) then maybe your deleted (why?) answer was the right one. $\endgroup$ – quim May 8 '13 at 14:41

1$\begingroup$ Ok, so let use the following definition that can be found in the paper by Iversen "Numerical invariants of multiple planes", p. 971. Let $x \in S$ and $y_1, \ldots, y_r$ be the points of $f^{1}x$. Let $d_i$ be the discriminant of the extension $\widehat{\mathcal{O}_x} \to \widehat{\mathcal{O}_{y_i}}$. Then a local equation of $D$ in $x$ is given by $\prod d_i$. $\endgroup$ – Francesco Polizzi May 8 '13 at 14:56
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The answer is yes when $\dim D=2$ and the variety upstairs (i.e. $C$ in your notation) is normal: see
Bas Edixhoven, Robin de Jong, Jan Schepers, Covers of surfaces with fixed branch locus, Lemma 2.1.
It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully.

$\begingroup$ My comments don't make sense any more, so I'll delete them in a moment. $\endgroup$ – quim May 9 '13 at 14:47