How/where are semi-log resolutions used? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:21:21Z http://mathoverflow.net/feeds/question/33095 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33095/how-where-are-semi-log-resolutions-used How/where are semi-log resolutions used? Franklin 2010-07-23T13:53:40Z 2010-11-01T04:51:34Z <p>In the <a href="http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.3592v1.pdf" rel="nofollow"><strong>paper</strong></a>, by János Kollár there is <em>problem 19</em> (page 8). It is one more strict resolution. A resolution that leaves untouched the semi-simple-normal-crossings singularities of pairs.</p> <p><strong>My question is:</strong> <em>How/where is that kind of resolution used/needed?</em></p> <p><strong>Quick definitions:</strong> </p> <p><em>Pair:</em> \$(X,D)\$ with \$X\$ algebraic variety and \$D\$ a Weil divisor on it.</p> <p><em>Semi-simple-normal-crossings:</em> 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\$. </p> http://mathoverflow.net/questions/33095/how-where-are-semi-log-resolutions-used/33156#33156 Answer by Zsolt Patakfalvi for How/where are semi-log resolutions used? Zsolt Patakfalvi 2010-07-24T00:43:03Z 2010-07-24T00:43:03Z <p>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.")</p> http://mathoverflow.net/questions/33095/how-where-are-semi-log-resolutions-used/42578#42578 Answer by Sándor Kovács for How/where are semi-log resolutions used? Sándor Kovács 2010-10-18T04:11:41Z 2010-11-01T04:51:34Z <p>Perhaps a little more explanation would be this:</p> <p>One of the first things we learn in algebraic geometry is <em>normalization</em> 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. </p> <p>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.</p> <p>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. </p> <p>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 <em>slc</em> is the non-normal version of <em>lc</em>).</p> <p>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.</p> <p>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).</p> <p>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.</p>