Suppose $f:X\rightarrow Y$ is a birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blowup 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.
1 Answer
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 $(Y, (I, 1), D)$.

$\begingroup$ Is it reasonable to expect the same to happen when $X$ and $Y$ are smooth schemes over some perfect field $k$ of positive characteristic? $\endgroup$ Commented Nov 24, 2023 at 18:49

$\begingroup$ Unfortunately, the proofs above only work because of resolution of singularities. We don't know these exist in positive characteristic I think (it is likely that this is ok for threefolds though  I'd want to double check the literature, see sciencedirect.com/science/article/pii/S0021869308001853 and sciencedirect.com/science/article/pii/S0021869308005942 and of course surfaces are fine). $\endgroup$ Commented Nov 26, 2023 at 23:35

$\begingroup$ Looking again, I think my situation is a bit different than the conjectured resolution of singularities. I have a birational morphism of smooth varieties $f:X_1 \to X_2$ agreeing on an open $U$, such that $D:=X_2 \backslash U$ is an snc divisor. I want to have a blowup sequance with smooth centers dominating $f$, such that the strict transform of $D$ intersects with the exceptional divisors AND is an isomorphism over $U$. $\endgroup$ Commented Nov 27, 2023 at 14:23

$\begingroup$ The standard approach, just pullback to $X_1$, will fail. I don't know if there are other results that would give it to you in characteristic $p > 0$. $\endgroup$ Commented Dec 3, 2023 at 18:46