Timeline for irreducible etale cover of a blowup
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Aug 29, 2014 at 0:56 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 22:29 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 22:23 | vote | accept | matthew | ||
| Aug 28, 2014 at 21:59 | answer | added | Will Sawin | timeline score: 1 | |
| Aug 28, 2014 at 18:33 | comment | added | Will Sawin | @matthew: Well the answer is not always for my contruction. | |
| Aug 28, 2014 at 18:11 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 17:54 | comment | added | matthew | @Will Sawin: The $X_i\times_{Y_i} X_i$ need not necessarily come from a base change $Y_i\rightarrow Y$ | |
| Aug 28, 2014 at 11:58 | comment | added | Will Sawin | @matthew: I didn't see that part of the question. The answer is not always. Take $X = \mathbb G_m$, $Y=\mathbb G_m$, with the map the squaring map. Then the fiber product is the disjoint union of two copies of the base, so any base change will be irreducible. | |
| Aug 28, 2014 at 7:10 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 7:05 | comment | added | matthew | @Laurent Moret-Just a set of indices corresponding to the cover morphisms. I'll edit the post to make it clearer | |
| Aug 28, 2014 at 6:50 | comment | added | Laurent Moret-Bailly | What does the index $i$ mean? | |
| Aug 28, 2014 at 4:51 | comment | added | matthew | @Will Savin:Why are the resulting schemes irreducible? | |
| Aug 28, 2014 at 3:00 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 2:56 | comment | added | Will Sawin | Can't one take $Y_i$ an etale cover of $Y$ and $X_i = X \times_Y Y_i$? Fiber product and blow-up are both etale-local constructions, so everything will just be a base change from $Y$ to $Y_i$, hence etale. | |
| Aug 28, 2014 at 2:00 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 1:36 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 0:10 | history | edited | matthew | CC BY-SA 3.0 |
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| Aug 28, 2014 at 0:07 | comment | added | matthew | Sorry, $W$ is the blowup of fiber product along the diagonal $W=Bl_{\Delta}X\times_Y X$, I'll edit the post. | |
| Aug 28, 2014 at 0:05 | comment | added | dhy | What are you denoting by $W$? | |
| Aug 27, 2014 at 23:15 | history | asked | matthew | CC BY-SA 3.0 |