Timeline for Is the Springer resolution a blow-up?
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11 events
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| Aug 7, 2013 at 20:27 | comment | added | JGordon | Thanks for a great question and all the comments. A related (naive) question: is anything known about a resolution of $\mathcal N$ via a sequence of blow-ups, and so that the exceptional divisor has normal crossings? | |
| Jul 12, 2012 at 15:07 | comment | added | user5117 | David: iirc another, maybe more familiar, example of the same phenomenon is the small resolution of the 3-fold ODP (as features in the Atiyah flop). Here there are non-Cartier Weil divisors through the singular point, and blowing up such a Weil divisor produces a small resolution. (Sorry for the off-topic comment.) | |
| Jul 12, 2012 at 14:04 | comment | added | David E Speyer | The result from Hartshorne which inkspot is referencing is Theorem II.7.17. So $\widetilde{\mathcal{N}}$ is a blow up of $\mathcal{N}$, on very general grounds, but it is not yet clear to me whether we can take it to be the blow up of a subscheme supported on the singular locus. | |
| Jul 12, 2012 at 14:00 | comment | added | David E Speyer | A subtlety which I hadn't realized before. Let $X$ be the subvariety $wy=wz=xy=xz=0$ of $\mathbb{A}^4$. This is the union of two coordinate $2$-planes, meeting transversely. Let $Y$ be the normalization of $X$, which separates the two planes. The map $Y \to X$ <b>is</b> a blowup. For example, it is the blowup at $\langle w,y \rangle$. However, note that the ideal $\langle w,y \rangle$ is not supported on the singular point of $X$; it is two crossing lines. The preimage of $\langle w,y \rangle$ is not the same thing as the exceptional locus. It is that preimage which is always Cartier. | |
| Jul 12, 2012 at 12:52 | comment | added | David E Speyer | @inkspot Doesn't the exceptional fiber have to be Cartier (and, in particular, codimension 1)? It is, in the case of the Springer resolution, but I'm not sure if that is a sufficient condition or only a necessary one. I'm pretty sure it isn't true that an arbitrary projective birational map is a blowup. | |
| Jul 12, 2012 at 12:46 | comment | added | inkspot | A projective birational morphism $X\to Y$ (e.g., the Springer resolution) is the blow-up of $Y$ along some (non-unique) subscheme; see Hartshorne. But it can be hard to identify such a subscheme. The Springer resolution is an example where nature provides a resolution where it is hard to find an appropriate subscheme. Others are: the Hilbert scheme of $0$-dimensional subschemes of a smooth surface, resolving the corresponding symmetric product; determinantal varieties; theta divisors on Jacobians; vector bundles over homogeneous spaces $G/P$ collapsing to some cone in a representation of $G$. | |
| Jul 12, 2012 at 11:55 | comment | added | Jim Humphreys | I'm also very skeptical. In rank 1 you just have a projective line as fiber at 0, but in higher Lie ranks the fibers get far more complicated geometrically. Anyway, it's important to look at more examples, say in rank 2 or 3, to avoid being misled. | |
| Jul 12, 2012 at 10:11 | comment | added | Bugs Bunny | Why is it not? I think it is not too but I cannot think of a reason why... | |
| Jul 12, 2012 at 6:12 | comment | added | Mike Skirvin | In the simplest case of $\mathfrak{sl}_2$, the Springer resolution is easily seen to be the blow-up at the origin. In general, however, the Springer resolution is not a blow-up. | |
| Jul 12, 2012 at 4:53 | history | edited | Zhaoting Wei | CC BY-SA 3.0 |
I have add a reference for Springer resolution.
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| Jul 12, 2012 at 4:13 | history | asked | Zhaoting Wei | CC BY-SA 3.0 |