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In the paper, by János Kollár there is problem 19 (page 8). It is one more strict resolution. A resolution that leaves untouched the semi-simple-normal-crossings singularities of pairs.

My question is: How/where is that kind of resolution used/needed?

Quick definitions:

Pair: $(X,D)$ with $X$ algebraic variety and $D$ a Weil divisor on it.

Semi-simple-normal-crossings: A point in $X$ where $X$ is (locally) a union of coordinates hyperplanes and $D$ is given by intersecting $X$ with some of the other coordinate hyperplanes not contained in $X$.

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By the way, the name is "János Kollár". –  Sándor Kovács Oct 18 '10 at 4:55
    
Fixed. Thank you. –  ABC Oct 19 '10 at 6:05
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2 Answers 2

up vote 16 down vote accepted

Perhaps a little more explanation would be this:

One of the first things we learn in algebraic geometry is normalization and we are told that it is "harmless" to assume that something is normal since the normalization exists and canonical and all that jazz. This is fine as long as one studies a stand alone object, but once they come in a family, it is no longer true.

Consider a family of curves. Recall that for curves normal=smooth so if not all members of the family are smooth, which is the likely scenario, then they are also not all normal. Now the problem is, there is no way to normalize the family members that they stay in a family. For instance, a family of smooth cubic plane curves degenerates to a singular one, but the normalization of the singular cubic has genus $0$, while the cubic curves have genus $1$ so they can't be members of the same family. Also, if you try to resolve the singularities by blowing up you'll see that you can resolve all singularities to be normal crossings, but you cannot do better and you also add new irreducible components to the singular fibers. This leads one to do semi-stable reduction, which is actually another story, so I won't get into that.

Anyway, for curves, we can actually make do with handling only smooth and simple normal crossing points. In higher dimensions if one tries to do the same, then there are other singularities that one must allow and these are the semi-log canonical (a.k.a. slc) singularities Zsolt mentioned.

OK, so maybe the above convinces you that if you want to do moduli theory and you want to study compact moduli spaces, that is, you actually would like to understand degenerations as well and not just the nice part, then you have to deal with non-normal, in particular with slc singularities. (Actually, "s-something" is usually the non-normal version of "something", accordingly slc is the non-normal version of lc).

Well, now how do you define a non-normal version of a singularity that is otherwise defined via some properties of exceptional divisors (or saying it in a more enlightened way: exceptional set)? You cannot take a full fledged resolution of singularities, because it will resolve the non-normality of the singularity as well. This would not be a huge problem from the point of view of making it simple, but it sabotages the entire operation. The issue is, that if you look at the definition of lc (and klt, dlt, etc) more closely, then it becomes clear that it kind of needs that the resolution used in the definition is an isomorphism in codimension $1$ on the target, that is, the singular guy. It is also important that the exceptional set is a divisor. These will fail for non-normal but $S_2$ singularities, for instance for slc but not lc singularities.

So, you need a partial resolution that resolves the singularities to something that is close to being smooth but has the above properties. The "close to being smooth" is called "semi-smooth", these are double normal crossings and pinch points, exactly the singularities that cannot be made better by only changing something in codimension $2$. (This last statement is left to the reader. If you have difficulty with it, ask).

OK, I better wrap this up. So the point of a semi-resolution is that it has those properties that make it possible to define discrepancies but it does not go "too far". However, it produces varieties with very mild singularities that are almost as good as smooth, at least from the point of view of this definition.

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Dear Sandor, I read it, and enjoyed it! –  Emerton Oct 18 '10 at 4:41
    
Hi Matt, thanks! :) –  Sándor Kovács Oct 18 '10 at 4:51
    
I will definitely read it and Emerton read it. I have a further question. You mentioned the importance of resolutions that are iso in codimension 1. This seems to be stronger than what Kolla'r is asking in problem 19. The question is: Is this correct? For example suppose the pair $(X,D)$ is $X:=(x_1x_2=0)$ and then $D:=(x_1=x_2=0)$. This pair is not semi-snc precisely along the support of $D$. Then to resolve it we need to blow-up $(x_1=x_2=0)$ and we don't get isomorphism in codimension 1. Is this correct? –  ABC Oct 19 '10 at 6:22
    
Franklin: I think Kollár does not say it, but assumes it. He does say it earlier: look at the bottom of page 5. Also, even though he does not explicitly say it, there are implicit assumptions in his theorems that rule out your example: See the definition of semi-snc in (10) and the assumptions on the restriction of $D$ in both (16) and (17). You are right, though, that in (19) he does not require it, but that is slightly different from needing it for the statement. Also, what I was saying is that you need this for the definition of slc, not that you would definitely need it for the... –  Sándor Kovács Oct 19 '10 at 7:12
    
...definition of a semi-log resolution. Anyway, in general, the sensible assumption to make about a pair is that none of the components of $D$ is contained in the singular locus of $X$. I suppose you could also weaken the requirement of the semi-log resolution from asking that it is an isomorphism in codimension $1$ (on $X$!) to that it is an isomorphism at general points of $D$. If you want to define slc, you need to be able to define the strict transform of $D$ and for that you need a piece of every component of it on the locus where $f$ is an isomorphism. –  Sándor Kovács Oct 19 '10 at 7:16
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It is used in the definition of semi-log canonical singularities (e.g. see Section 4 of "Kollár, J.; Shepherd-Barron, N. I. Threefolds and deformations of surface singularities. Invent. Math. 91 (1988), no. 2, 299--338.")

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Thank you very much. –  ABC Jul 24 '10 at 2:15
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