Timeline for Smooth projective varieties of Picard number one
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 7, 2013 at 3:32 | vote | accept | Fei YE | ||
| Mar 1, 2013 at 8:21 | answer | added | Sándor Kovács | timeline score: 7 | |
| Feb 26, 2013 at 11:25 | comment | added | Chris Brav | As for classification, in general no. Sometimes if you fix some additional geometric conditions, say smooth Fano threefold, then there is a classification of those with Picard number 1. But for other geometric conditions, say Calabi-Yau threefolds, then (last time I checked) there is not yet a classification of those with Picard number 1. | |
| Feb 26, 2013 at 10:31 | answer | added | Yuri Zarhin | timeline score: 11 | |
| Feb 26, 2013 at 8:54 | comment | added | Serge Lvovski | There are many. Not only most Grassmannians, but most flag varieties (of semisimple groups) with Picard number one are not complete intersections. A still bigger series of examples is as follows. Consider an arbitrary smooth projective variety $X\subset\mathbb P^n$ with Picard number one. Even if it is a complete intersection in $\mathbb P^n$, its $m$-fold embedding, for $m\gg0$, is not a complete intersection anymore (proof: canonical class of a c.i. must be a multiple of the class of hyperplane section). | |
| Feb 26, 2013 at 8:29 | comment | added | Fei YE | Thanks a lot for the comment. Are there other types? | |
| Feb 26, 2013 at 6:24 | comment | added | Atsushi Kanazawa | For example, Grassmannian $Gr(k,n)$ is in general not complete intersection of any projective spaces, but is smooth projective and has Picard number $1$. | |
| Feb 26, 2013 at 5:44 | history | asked | Fei YE | CC BY-SA 3.0 |