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Suppose we have a collection, $S$, of singularities types and consider a resolution of singularities (this is: a proper birrational morphism $Y\rightarrow X$ such that Y only contains singularities of the types in the collection $S$ and such that the map is an isomorphism over the points of $X$ with singularity types in $S$.).

For example, if $S$ consists only of the smooth points then $S$-strict resolution is just the standard resolution of singularities. If $S$ consists of smooth points and simple normal crossing points $S$-strict resolution exists. If $S$ consists of only the smooth points and normal crossings points then there is no $S$-strict resolution (in embedding dimension at least 3). As you can't resolve a pinch piont $(x^2+yz^2=0)$ without blowing up normal crossing points.

Have you come around situations in which it is needed strict resolutions (excluding the cases of the first two examples)?

Extension of question (I guess this should go here and no in a new question) It is exactly the same question but this time asking for resolutions that never involve blowing up a center that intersects the S-locus of the total transform.

eg: -)For S= the smooth points the usual resolution is still fine since it never need to blow-up smooth points. If S is the smooth points and the simple normal crossings singularities this is already not known (I think).

Again the question is: Have you come around situations in which it is needed strict resolutions in this sense (excluding the first example in this second part)?

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3 Answers

For S = {rational singularities} (assuming I've understood the statement right) this is an open and presumably hard question. It's something that Koll\'ar wanted (see Chapter 12, I think, of Kollar's Shaferavich Maps and Automorphic forms). Essentially this would let you know that if $X$ has rational singularities, then it also has a compactification $\bar{X}$ with rational singularities (which is what Kollar wanted it for I think).

On the other hand, if you are doing more MMP type singularities (for example, S = log terminal singularities), then this kind of thing is often possible due to the MMP. See for example BCHM, Corollary 1.4.3 and also KK, Theorem 3.1. (I've been told more general statements exist also, but I don't know a reference).

EDIT1: For S = Cohen-Macaulay, this is also an open question (Macaulayfication is not strict).

EDIT2: Also see the literature on semi-resolutions. While this isn't quite the same thing, it's very closely related (the idea there is to leave the codim 1 singularities alone if they are nice enough, ie SNC or pinch points). This is a fundamental construction in the study of moduli spaces of higher dimensional varities.

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In terms of why it was wanted for rational singularities, Kollar wanted to prove that rational singularities were Du Bois, and showed that rational compactification would be enough to do it. Later, Kovacs and Saito independently proved that rational => Du Bois without this. –  Karl Schwede Jul 12 '10 at 14:20
    
In terms of the MMP stuff, this has been used in many many places. In the KK citation (which actually cites a theorem due to Hacon), they are using it to show that log canonical singularities are Du Bois. The reason for this is understanding the moduli spaces of higher dimensional varieties. –  Karl Schwede Jul 12 '10 at 14:30
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I have seen a resolution which is shown to be $S$-strict constructed for the purpose of determining the $S$ locus of $X$.

Suppose you have a subvariety $Z\subset X$ of points which you know (say by explicit computation) to be non-$S$, and you conjecture this is the non-$S$ locus of $X$. One way of proving this conjecture is to construct a resolution $Y\rightarrow X$ such that $Y$ is $S$ and the exceptional locus is (contained in) $Z$. It follows from the assumption in the first sentence that the resolution is $S$-strict.

Perrin carried this out for $S$="Gorenstein" and $X$="any Schubert variety of G/P for P a minuscule parabolic" in "The Gorenstein locus of minuscule Schubert varieties", Adv. Math. 220 (2009), no. 2, 505--522.

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Just to remark a typo: it should be "the exceptional locus is (contained in) the preimage of $Z$". –  Sándor Kovács Oct 19 '10 at 7:53
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And here's the most obvious one:

S = Normal singularities

One can of course also include seminormal and weakly normal.

EDIT: You can also add S2 (and thus the operation of S2-ification). Note that all of these have \emph{finite} "resolutions".

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Thanks, for these additions. Can you point out to where are they needed or why are they needed? –  ABC Jul 12 '10 at 13:29
    
For normality, it's very useful to know that normalization is an isomorphism on the normal locus. This is ubiquitous and very basic. For example, it appears in the study of integral closure of ideals (see the book of Huneke and Swanson) and also appears in various questions in algebraic geometry when dealing with non-normal things (see for example the paper of Kollar and Shepherd Barron). –  Karl Schwede Jul 12 '10 at 14:33
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