Timeline for Subvarieties with isomorphic complements
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2019 at 3:53 | comment | added | Zhiyu | @R.vanDobbendeBruyn Thanks, so the case $X$ is a surface is also true as stably birational smooth projective curves must be isomorphic. For the case $X$ is a threefold, is there any restriction for two smooth projective surfaces to be embedded in a common threefold ? | |
| Oct 5, 2019 at 7:11 | comment | added | R. van Dobben de Bruyn | It implies $[Y_1] = [Y_2] \in K_0(\mathbf{Var}_{\mathbf C})$. For example, this implies (in the smooth projective case) that $H^{p,q}(Y_1) = H^{p,q}(Y_2)$ for all $p,q$ (Deligne) and that $Y_1$ is stably birational to $Y_2$ (Larsen–Lunts). A first example of non-isomorphic smooth projective varieties with the same class in $K_0$ is $\mathbf P^1 \times \mathbf P^1$ and $\operatorname{Bl}_p \mathbf P^2$. I don't know how to make these occur in an $X$ with isomorphic complement, but maybe people who understand toric varieties can easily write down such a thing. | |
| Oct 5, 2019 at 3:04 | history | edited | Zhiyu | CC BY-SA 4.0 |
added 5 characters in body; edited tags
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| Oct 4, 2019 at 20:46 | history | asked | Zhiyu | CC BY-SA 4.0 |