Timeline for Trouble with semicontinuity
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Mar 8, 2012 at 1:35 | comment | added | Jack Huizenga | Playing around with this in Macaulay, I'm skeptical that $x_2$ is actually a nilpotent. Shouldn't the special fiber just be the reduced quartic curve? The ideal choa provided for the special fiber is not saturated, and does not generate the ideal in affine charts. One way to "fix" the exercise would be to look at $H^i(t,I_t(1))$ for $i=0,1$; then its just the statement that the special fiber is contained in a plane. | |
| Mar 7, 2012 at 23:57 | vote | accept | Choa | ||
| Apr 25, 2012 at 2:59 | |||||
| Mar 7, 2012 at 4:03 | comment | added | Sándor Kovács | choa, $x_2$ is a nilpotent, so it is contained in every prime ideal, so as a function it will always be zero. $x_0$ is not a section of $\mathscr O_{\mathscr X_0}$, so you can't divide with it. | |
| Mar 7, 2012 at 4:01 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Mar 7, 2012 at 1:36 | vote | accept | Choa | ||
| Mar 7, 2012 at 2:27 | |||||
| Mar 7, 2012 at 1:20 | comment | added | Choa | Thanks for yourntrue kindness and spending your time for such a tedious calculation to check everything for me:) Thank you very much. | |
| Mar 7, 2012 at 0:12 | vote | accept | Choa | ||
| Mar 7, 2012 at 1:35 | |||||
| Mar 6, 2012 at 18:02 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Mar 6, 2012 at 3:32 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Mar 6, 2012 at 2:22 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Mar 6, 2012 at 2:21 | comment | added | Sándor Kovács | Dear choa, the thing is, that this $x_2$ is a renegade on $X_0$, because $X_0$ is inside $x_2=0$. If you look at affine charts on $X_0$, you see that $x_2$ survives in all of them, so it glues to give a global section. On the entire $\mathbb P^4$ it does not give a section on the affine charts either, but in the $\mathbb P^3$ defined by $x_2=0$ it is independent of the variables. | |
| Mar 6, 2012 at 2:06 | vote | accept | Choa | ||
| Mar 6, 2012 at 8:47 | |||||
| Mar 6, 2012 at 0:48 | comment | added | Choa | When we take $Proj$ operation, cutting out lower degree parts changes nothing, so I ignored $x_{2}$...... | |
| Mar 6, 2012 at 0:44 | comment | added | Choa | Thank you so much! One more question; is that nilpotent function are really "global"? I mean, when I considered that $x_{2}$, I concluded that it cannot be captured as a global section. | |
| Mar 5, 2012 at 23:36 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Mar 5, 2012 at 23:36 | history | undeleted | Sándor Kovács | ||
| Mar 5, 2012 at 23:35 | history | deleted | Sándor Kovács | ||
| Mar 5, 2012 at 23:00 | history | answered | Sándor Kovács | CC BY-SA 3.0 |