Timeline for Nice small resolution and normality of blow-up
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2021 at 3:10 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 16 characters in body
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| Mar 29, 2021 at 16:22 | history | edited | pi_1 |
edited tags
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| Mar 29, 2021 at 13:25 | comment | added | pi_1 | Thank you very much for your remark. I have corrected the question. | |
| Mar 29, 2021 at 13:20 | history | edited | pi_1 | CC BY-SA 4.0 |
The example provided by Tom Ducat shows that the $Bl_Z(X)$ is not smooth in general
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| Mar 29, 2021 at 13:11 | comment | added | Tom Ducat | I am not exactly sure how you intend to apply Zariski's main theorem, but the bit in parentheses on the last line cannot be true. If $X_n$ is the isolated 3-fold hypersurface singularity $X_n=(xy = z(z+t^{n}))$ in $\mathbb A^4_{x,y,z,t}$ where $n\geq2$, then $Z$ is a point, and it admits a small resolution with $f^{-1}(Z)\cong \mathbb P^1$ by blowing up the ideal $(x,z)$ (this is one side of a 3-fold flop known as Reid's pagoda). The blowup $Bl_ZX_n$ is normal, but not smooth. | |
| Mar 29, 2021 at 10:23 | history | asked | pi_1 | CC BY-SA 4.0 |