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I've been doing some very messy computations with normalizations of various surfaces, and I really want to not have to do them.

So my question is this: Let $S$ be a dimension $2$ integral scheme (you can pretend it's a variety, or that it's affine, or whatever helps you), which is singular along an irreducible divisor $C$ -- can we tell how many divisors lie above $C$ in the normalization of $S$, without computing the normalization of $S$?

This question is not very well defined - but what I'm looking for is an answer that is easier than to compute than the normalization of $S$.

This question can be asked in any dimension, but I'm particularly curious about the dimension $2$ case. In dimension $1$ (at least for plane curves) I think this is classic and known, but I can't find a reference.

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    $\begingroup$ Dear Makhalan: The problem can be localized at the generic point of the divisor $C$, so it is really the question of computing the number of maximal ideals in the normalization of the 1-dimensional local ring of the surface $S$ at the generic point of $C$. In this respect, the underlying algebraic problem is exactly the same as that of computing the fiber under normalization of a non-regular point on an algebraic curve. And it can be computed in exactly the same way: by successive blow-up of closed points (over a 1-dimensional local domain). Nothing special about dimension 2 here... $\endgroup$
    – BCnrd
    Commented Oct 5, 2010 at 0:17
  • $\begingroup$ Just to be clear, the general fact is that if $A$ is any 1-dimensional semi-local noetherian domain whose normalization $A'$ is module-finite over $A$, then $A'$ is obtained in finitely many steps by repeatedly blowing up closed points at which the local ring isn't a dvr. In this special setup, the blow-up is always module-finite. Computing blow-ups in general may be a pain in the neck (1-dim. singularities can be nasty), but certainly not worse than trying to compute normalizations in one fell swoop. Then again, I hardly ever compute anything, so maybe someone else will have a better idea. $\endgroup$
    – BCnrd
    Commented Oct 5, 2010 at 0:26
  • $\begingroup$ Thanks, Brian. So let me sure I've got this right. If $A$ is a ring, and $\mathfrak{p}$ is a prime ideal there, and $\mathfrak{q}_1,...,\mathfrak{q}_r$ are the prime ideals lying above $\mathfrak{p}$ in $\tilde{A}$ (the normalization) then $\widetilde{A_{\mathfrak{p}}}$ would be iso. to $\bigoplus\tilde{A}_{\mathfrak{q}_i}$? $\endgroup$
    – Makhalan Duff
    Commented Oct 5, 2010 at 0:59
  • $\begingroup$ Dear Makhalan: I suppose your $A$ is a domain whose normalization is module-finite. Normalization of $A_{\mathfrak{p}}$ is semi-local with its local rings at closed points equal to the localizations of $\widetilde{A}$ at the $\mathfrak{q}_i$. Consider the natural map $\widetilde{A_{\mathfrak{p}}} \rightarrow \prod \widetilde{A}_{\mathfrak{q}_i}$ (please do not write $\oplus$ with rings, use $\prod$; categorically it matters). This is typically not an isom: $\widetilde{A_{\mathfrak{p}}}$ is a domain, after all! Even for alg. curves this fails. The etale topology rescues analytic intuition... $\endgroup$
    – BCnrd
    Commented Oct 5, 2010 at 1:42
  • $\begingroup$ Thanks again, Brian. That's what I meant, but I was just confused. Of course, as was written, my statement was in general false. I too push people to write products when it is the categorical product, but alas, I slipped. $\endgroup$
    – Makhalan Duff
    Commented Oct 5, 2010 at 14:57

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While what Brian wrote above is certainly correct, you can sometimes estimate the number of components without computing the normalization. For example, localize at the height one prime as described above. Then I think the multiplicity (with respect to the maximal ideal) of that non-normal 1-dimensional domain of will give you an upper bound on the number of divisors lying over your given divisor. This might be substantially harder than computing the resolution though, depending on what you know about your example.

If you know additional things about that 1-dimensional domain, like for example, it is Gorenstein and/or seminormal, then one should be able to make even more precise statements. For example, blowing up the conductor can under certain circumstances be shown to compute the normalization (see Greco-Traverso). Alternately, if the ring is seminormal, then the embedding dimension should give you an upper bound on the number of divisors lying over the point.

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