Timeline for Smooth normalization and blow-up of the exceptional locus
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 20, 2021 at 19:21 | comment | added | Jason Starr | You can easily modify my example to be connected: simply take the product with some smooth variety $Y$ that admits a finite, unbranched, 2-to-1 cover, $Y'\to Y$. My example variety $X$ admits an involution that permutes the two branches at the tacnode, namely $(s,t,0,0) \leftrightarrow (s,t,s^2,t^2)$. Form the quotient of $X\times Y'$ by the "diagonal" involution. Then the singular is connected, isomorphic to $Y$, and the normalization is isomorphic to the quotient of $\widetilde{X}\times Y'$ by the diagonal involution. So the inverse image of the singular locus is isomorphic to $Y'$. | |
| May 4, 2021 at 18:32 | comment | added | pi_1 | Thanks. The codimension $>1$ does not add anything: it does not guarantee that the projectivized normal is disconnected. | |
| May 4, 2021 at 18:16 | comment | added | pi_1 | Thank you very much for your counter-example. I edited the question to be closer to the situation I am interested in. | |
| May 4, 2021 at 18:15 | comment | added | Jason Starr | Oops! A tacnodal curve fails the "codimension $>1$" hypothesis for $Y$. Instead, consider a "tacnodal surface", e.g., the zero scheme in affine $4$-space $\text{Spec}\ \mathbb{C}[s,t,u,v]$ of the ideal $\langle u,v\rangle \cap \langle u-s^2,v-t^2 \rangle$. | |
| May 4, 2021 at 17:47 | history | edited | pi_1 | CC BY-SA 4.0 |
added 30 characters in body
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| May 4, 2021 at 17:33 | comment | added | Jason Starr | That fails if $X$ is a tacnodal curve. | |
| May 4, 2021 at 10:26 | history | asked | pi_1 | CC BY-SA 4.0 |