Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
1  
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. –  Joël May 8 '13 at 13:10
2  
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. –  Francesco Polizzi May 8 '13 at 13:41
    
Ah okay. So what is the definition of the schematic branch locus? –  Joël May 8 '13 at 14:41
    
@Francesco: We should be able to see that the schematic branch locus has multiplicity two by looking at the ring map $\mathbb{C}[x]\rightarrow\mathbb{C}[x,y]/(x^2-y^2)$. 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. –  quim May 8 '13 at 14:41
1  
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 $\prod d_i$. –  Francesco Polizzi May 8 '13 at 14:56
show 1 more comment

1 Answer

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.

share|improve this answer
    
My comments don't make sense any more, so I'll delete them in a moment. –  quim May 9 '13 at 14:47
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.