Timeline for Normality via resolution of singularities
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2011 at 15:28 | comment | added | Alexander Braverman | Sorry I found your comment only now - I actually won't be at Brown today, and I guess you are not staying till tomorrow... | |
| Sep 30, 2011 at 23:36 | comment | added | Jason Starr | You might be able to say something about the singularities by considering an incidence correspondence of (projectively equivalent) embeddings of the flag variety into projective space. I did something like this in the following paper. The Kodaira dimension of spaces of rational curves on low degree hypersurfaces arxiv.org/abs/math/0305432 If you would like to hear more, I could meet with you on Monday (I will be at Brown on Monday). | |
| Sep 30, 2011 at 14:37 | comment | added | Alexander Braverman | I meant to say "we know everything about the canonical class of the resolution". | |
| Sep 30, 2011 at 14:35 | comment | added | Alexander Braverman | Well, the situation is this: I am looking at quasi-maps from ${\mathbb P^1}$ to the flag variety $X$ of a semi-simple group $G$. It has a resolution by of stable maps into $X\times {\mathbb P}^1$ which have degree 1 over ${\mathbb P}^1$. The fibers of this map are easy to describe (even scheme-theoretically). If $X$ is ${\mathbb P}^n$ then the space of quasi-maps is smooth and I also know normality when $G=SL(n)$ (by different means). Also, we know everything about the resolution - you can use this to show that normality implies rational singularities - I don't know if it helps. | |
| Sep 30, 2011 at 13:01 | comment | added | Jason Starr | I don't think there can be a criterion based only on the dimension of the non-reduced locus. To expand on Steven's example, you can take the origin in affine n-space and put a "cusp", i.e., non-normal point, only at that point. The resolution will be a homeomorphism with domain equal to affine n-space, and the only non-reduced fiber will be the origin. But of course there could be some result in your situation. Can you tell us more about your situation? | |
| Sep 30, 2011 at 11:31 | comment | added | Alexander Braverman | My question was not formulated precisely - what I had in mind was for example whether some careful analysis of dimensions on the non-reduced locus compared to some other parameters of the situation can help; or maybe a group action with some properties. Basically I am wondering if there are examples when one can prove normality based on some purely geometric properties of the situation. | |
| Sep 30, 2011 at 5:54 | comment | added | rita | I think the question is interesting only for dimension >1, since for curves normalization and desingularization coincide. | |
| Sep 30, 2011 at 5:44 | comment | added | Steven Landsburg | Alexander: Have I misundersood your question? I thought you were asking "If the fibers of the map $\pi$ are not reduced but nevertheless very nice, can I conclude that $X$ is normal?". I gave an example of a map where the fibers are not reduced but nevertheless as nice as non-reduced fibers can possibly be, yet $X$ is not normal. So if I understood your question correctly, I believe this pretty much answers it. | |
| Sep 30, 2011 at 4:53 | comment | added | Alexander Braverman | It is obvious that if the fibers are reduced and connected then the variety is normal. In your case the fibers are obviously not reduced. | |
| Sep 30, 2011 at 4:10 | history | answered | Steven Landsburg | CC BY-SA 3.0 |