Timeline for Blowups of Cohen-Macaulay varieties
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2022 at 13:03 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Oct 24, 2010 at 14:39 | comment | added | Sándor Kovács | Karl, yes, regarding the map in #4, these are exactly my worries, but here is what I was thinking: It seems that the morphism is OK outside of the vertices of the cones, so if you resolve the indeterminacies, then you get a morphism extending this on a scheme that you get by blowing up an ideal supported at the vertex of $Y$ and the morphism maps the entire exceptional set to the vertex of $X$. But then, it seems that this morphism should actually factor through $Y$. In fact, this might be a place where $Y$ being normal makes a difference. | |
| Oct 24, 2010 at 14:01 | comment | added | Karl Schwede | S\'andor, I agree that it seems unlikely to be true. For 4), I'm slightly confused, because $Y$, the cone over $Z$ is also affine, so I don't think the map $f:Y\to X$ will be proper+birational (is there even a map in general). For the existence of $f$, I suppose it depends on the choice of a particular line bundle on $Y$ to take the sectionf. Start with the pull back of whatever ample on $X$ twisted by something relatively ample? My guess is that the map $Y \to X$ will not be defined everywhere, maybe resolving the indeterminacies will help... then do what you did? I'll think about this. | |
| Oct 24, 2010 at 5:46 | history | edited | Sándor Kovács | CC BY-SA 2.5 |
Changed #6 to include more stuff
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| Oct 24, 2010 at 5:27 | history | answered | Sándor Kovács | CC BY-SA 2.5 |