Timeline for Ample toric divisors
Current License: CC BY-SA 4.0
17 events
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| S Jan 28, 2023 at 20:07 | history | bounty ended | CommunityBot | ||
| S Jan 28, 2023 at 20:07 | history | notice removed | CommunityBot | ||
| Jan 24, 2023 at 2:55 | comment | added | Mohammad Farajzadeh-Tehrani | Kollár says that, if $dim X\geq 3$, this is true by Grothendieck-Lefschetz theorem, but is there a version of this theorem for normal crossings $D$? Where can I find it. | |
| Jan 23, 2023 at 20:07 | comment | added | Nick L | This is not a counter-example, but seems to be non-toric in a quite strong sense. Consider a line and a smooth conic in $\mathbb{P}^2$ intersecting it transversally in two points. Then this toric arrangement embeds equivariantly for some $\mathbb{C}^*$-action on $\mathbb{P}^2$ but is not a toric divisor. It also shows that $\pi_{1}$ of the complement can vary for different configurations (i.e. $\mathbb{Z}^2$ for 3 lines or $\mathbb{Z} \times \mathbb{Z}_2$ for the above one), maybe this has some consequence for Lefschetz type arguments (or maybe not). It is however, log Calabi-Yau. | |
| Jan 23, 2023 at 16:11 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |
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| Jan 23, 2023 at 15:54 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |
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| S Jan 20, 2023 at 18:33 | history | bounty started | Mohammad Farajzadeh-Tehrani | ||
| S Jan 20, 2023 at 18:33 | history | notice added | Mohammad Farajzadeh-Tehrani | Draw attention | |
| Jan 19, 2023 at 18:11 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jan 19, 2023 at 15:53 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |
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| Jan 18, 2023 at 23:13 | comment | added | Jason Starr | You are right. I will try to think of some higher-dimension examples | |
| Jan 18, 2023 at 18:52 | comment | added | Mohammad Farajzadeh-Tehrani | That does not meet the toric condition on $D_i$! | |
| Jan 18, 2023 at 17:22 | comment | added | Jason Starr | There are higher-dimensional examples: let $X_1,\dots,X_d$ be hyperbolic curves, let $D_1,\dots,D_r$ be singleton divisors in each $X_i$, let $X$ be $X_1\times \dots \times X_r$, and let $D$ be the sum of the pullbacks of the $D_i$ by the coordinate projections. | |
| Jan 18, 2023 at 13:36 | comment | added | Mohammad Farajzadeh-Tehrani | In the context I was thinking about, $X-D$ has hyperkahler metric, but I doubt the last condition is really important. | |
| Jan 18, 2023 at 13:27 | comment | added | Mohammad Farajzadeh-Tehrani | Are there higher dimensional examples? | |
| Jan 18, 2023 at 12:27 | comment | added | Jason Starr | What if $X$ is a hyperbolic curve and $D$ is a single point? | |
| Jan 18, 2023 at 3:30 | history | asked | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |