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Suppose that $C$ and $D$ are curves of finite type over an algebraically closed field $k$ (we make some of these hypotheses for simplicity). We view these as pointed curves with singularities $c \in C$ and $d \in D$. I'm happy to assume that these singularities look analytically locally like $n$-coordinate-lines through the origin in $\mathbb{A}^n$; in other words $k[x_1, \dots, x_n]/(\dots, x_i x_j, \dots)$.

If you'd like to start with simple (planar) nodes (ie, $k[x,y]/(xy)$), that's ok too.

Further suppose that there is a finite map $C \to D$ taking $c$ to $d$.

Question: Is there an accepted notion of what it means for such a map to have tame ramification at $d$?

Particularly, has it been studied before, and if so, where can I read about it?

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2 Answers 2

I have no idea if there's an accepted notion, but in case you're also interested in possible definitions, here's one from logarithmic algebraic geometry. When $C$ and $D$ are smooth curves, being tame at $c$ is equivalent to being log-etale, where the log structures on $C$ and $D$ are the ones inherited from the trivial log structures on $C-\{c\}$ and $D-\{d\}$. (At least, that's how I remember it. It's been a while.) So it seems to me that the same definition would still be a reasonable candidate without any smoothness assumptions, though of course everything depends on what you want to do with it.

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Thanks, that's a good idea and one I hadn't thought of. In the log smooth setting, we're only going to be looking at nodes (2 points glued), right? It's been awhile since I thought about log geometry. –  Karl Schwede Sep 21 '10 at 2:11

Here is a reference that may be relevant (you may know it already): Auslander-Rim has a paper called "Ramification index and multiplicity" . Even though they mostly discussed normal rings, their definition of tame ramification used only Hilbert-Samuel multiplicity, so it can be adapted to non-normal situations.

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Long, I meant to mention this to you a while ago, but forgot. The ramification index they are discussing seems to be a different one than the modern one (at least in the wildly ramified setting). –  Karl Schwede Nov 21 '10 at 17:06
    
@Karl: that's too bad, sorry it didn't help. –  Hailong Dao Nov 21 '10 at 18:40

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