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Suppose $f:X\rightarrow Y$ be an birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blow ups sequence on $Y$ which dominates $f$ such that the preimage of D and all other subsequent preimages involved are snc divisors? Thanks for any comments or references.

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up vote 7 down vote accepted

Here's an easy way to do it for morphisms in characteristic zero.

Suppose that $f : X \to Y$ is projective and thus the blowup of some ideal $I$ on $Y$, see Harthsorne Chapter II, Section 7 (if $f$ is not projective, make it projective by compactifying and using Chow's lemma).

The nice thing about setting things up this way is that now if $Z \to Y$ is any log resolution / principalization of $(Y, I)$, then $Z \to Y$ factors through $X$ by the universal property of blowing up.

Goal: All you need to do is a sequence of blowups $h : Z \to Y$ at smooth centers on $Y$ that principalizes $I$ to become an SNC divisor $E$ and keeps $h^{\star} D \cup E$ SNC (and keeps the pullback of $D$ SNC at each stage)

Fortunately, modern resolution algorithms do exactly this if I recall correctly. Indeed, when you are run a modern resolution algorithm, you are running it somewhat recursively on data $(I, D)$ where $D$ is a SNC divisor obtained by previous blowups (you need to keep track of this data for the resolution algorithm to work at further steps).

In particular, if you pass your resolution algorithm both $D$ and $I$ (in other words, telling the algorithm that this divisor needs to be handled with care), you should be able to do exactly what you want.

I would suggest checking out the following sources to see if what I said is actually accurate :-) (It's been a long time since I thought seriously about this).

Explicitly, in reference 2 below, you should apply Theorem 3.10 to the basic object $(Y, (I, 1), D)$.

  1. János Kollár's book

  2. A paper by Bravo-Encinas-Villamayor

  3. A paper by Wlodarczyk

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Thanks a lot. You are very nice. –  MZWang Sep 5 '12 at 2:27
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