Timeline for Does smooth target space and smooth fibers imply smooth total space?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2010 at 19:02 | comment | added | Karl Schwede | Yes, I do mean $(a^2*c - b^2)$, thanks. Yes, the total is singular along every point of the exceptional divisor, let me think about the other question (but again, it might be better as a separate or modification of the original question as David said). | |
| Jun 1, 2010 at 18:28 | comment | added | David E Speyer | @unkown: You'll probably get better responses if you pose you ask this question as a separate question, not a comment. | |
| Jun 1, 2010 at 15:59 | comment | added | unknown | The example is cool, the total space is singular at every point along the exceptional divisor right? The singularities are not detected over the closed point - but they would be if we look at some neighborhood, like k[a,b]/m^3 where m = (a,b) the maximal ideal. If X \rightarrow Y, and x \in X, y = f(x), if the fiber is smooth over every infinitessimal neighborhood of y, is X smooth at x? | |
| Jun 1, 2010 at 15:33 | comment | added | unknown | Ah, I think you are right (though you mean b^2 in your formula). Thanks, is there any other condition on the situation (besides flatness) which would guarantee that the domain is smooth or is it just hopeless? | |
| Jun 1, 2010 at 15:15 | history | edited | Donu Arapura | CC BY-SA 2.5 |
added 147 characters in body; added 80 characters in body
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| Jun 1, 2010 at 15:11 | comment | added | Karl Schwede | Modifying Donu's example very slightly, blow up the ideal $(x^2, y^2)$. You have two charts, one is $k[x, y, (y/x)^2] = k[a,b,c]/(a^2*c - b)$ and the other is the other obvious (and symmetric) one. This chart isn't smooth, the fiber over the closed point $(x, y)$ is $k[c]$. | |
| Jun 1, 2010 at 15:06 | comment | added | Donu Arapura | OK, sorry. I misread your question. | |
| Jun 1, 2010 at 15:00 | comment | added | unknown | I'm asking about the domain X, not the morphism f. | |
| Jun 1, 2010 at 14:41 | comment | added | unknown | The blowup of a smooth variety along a smooth subvariety is still smooth though? | |
| Jun 1, 2010 at 14:35 | history | answered | Donu Arapura | CC BY-SA 2.5 |