Timeline for Total space of the line bundle $\mathcal{O}(1)$ over $\mathbb{P}^n$
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 7, 2016 at 12:13 | comment | added | Tony Pantev | $Tot(\mathcal{O}_{\mathbb{P}^{n}}(k))$ is $\mathbb{P}^{n+1}(1,1,\ldots,1,k) - \{x\}$, where $x = (0:0:\ldots:0:1)$. | |
| Apr 5, 2016 at 1:45 | comment | added | PrimeRibeyeDeal | @TonyPantev would you mind saying which weighted projective space contains the total space of $\mathcal{O}(n)$ and how to think of it as a toric variety? | |
| Nov 9, 2010 at 1:22 | comment | added | Tony Pantev | You can get $Tot(\mathcal{O}(n))$ in a similar manner by deleting a point from a weighted projective space. But this is more contrived. And is not really better than thinking of $Tot(\mathcal{O}(n))$ as a toric variety. So this probably is not what you want. | |
| Nov 8, 2010 at 14:23 | comment | added | Klim Puhov | Thank you, Tony. Maybe you know such simple description of $Tot(O(n))$ for $n>1$ also? | |
| Nov 7, 2010 at 20:19 | vote | accept | Klim Puhov | ||
| Nov 7, 2010 at 20:17 | vote | accept | Klim Puhov | ||
| Nov 7, 2010 at 20:17 | |||||
| Nov 7, 2010 at 2:20 | comment | added | Tony Pantev | The tautological line bundle contains a divisor, isomorphic to $\mathbb{P}^n$, with normal bundle of degree $-1$ (namely, the zero section). On the other hand every effective compact divisor in $\mathbb{P}^{n+1}-\{x\}$ has a positive normal bundle. In particular, a hyperplane avoiding $x$ has normal bundle of degree $1$. The projection from the point identifies the complement of $x$ with the total space of this normal bundle. | |
| Nov 7, 2010 at 1:59 | comment | added | Fei YE | Shouldn't $\mathbb{P}^{n+1}\setminus\{x\}\to \mathbb{P}^n$ be the tautological line bundle? | |
| Nov 6, 2010 at 23:59 | history | answered | Tony Pantev | CC BY-SA 2.5 |