What strict resolutions of singularities are needed? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:21:36Z http://mathoverflow.net/feeds/question/31466 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31466/what-strict-resolutions-of-singularities-are-needed What strict resolutions of singularities are needed? Franklin 2010-07-11T21:19:46Z 2010-07-12T15:00:59Z <p>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 $S$.). </p> <p>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. </p> <p>Have you come around situations in which it is needed strict resolutions (excluding the cases of the first two examples)? </p> <p><strong>Extension of question</strong> (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. </p> <p>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).</p> <p>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)?</p> http://mathoverflow.net/questions/31466/what-strict-resolutions-of-singularities-are-needed/31488#31488 Answer by Alexander Woo for What strict resolutions of singularities are needed? Alexander Woo 2010-07-12T01:09:14Z 2010-07-12T01:09:14Z <p>I have seen a resolution which is shown to be $S$-strict constructed for the purpose of determining the $S$ locus of $X$.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/31466/what-strict-resolutions-of-singularities-are-needed/31500#31500 Answer by Karl Schwede for What strict resolutions of singularities are needed? Karl Schwede 2010-07-12T04:06:20Z 2010-07-12T14:32:34Z <p>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).</p> <p>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 <a href="http://front.math.ucdavis.edu/0610.5203" rel="nofollow">BCHM</a>, Corollary 1.4.3 and also <a href="http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.0648v4.pdf" rel="nofollow">KK</a>, Theorem 3.1. (I've been told more general statements exist also, but I don't know a reference).</p> <p>EDIT1: For S = Cohen-Macaulay, this is also an open question (Macaulayfication is not strict). </p> <p>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.</p> http://mathoverflow.net/questions/31466/what-strict-resolutions-of-singularities-are-needed/31543#31543 Answer by Karl Schwede for What strict resolutions of singularities are needed? Karl Schwede 2010-07-12T12:46:54Z 2010-07-12T14:29:06Z <p>And here's the most obvious one:</p> <p>S = Normal singularities</p> <p>One can of course also include seminormal and weakly normal. </p> <p>EDIT: You can also add S2 (and thus the operation of S2-ification). Note that all of these have \emph{finite} "resolutions".</p>