Timeline for Cone over the Veronese surface
Current License: CC BY-SA 4.0
10 events
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| Nov 16, 2020 at 23:56 | history | edited | Sándor Kovács | CC BY-SA 4.0 |
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| Jan 8, 2015 at 6:57 | comment | added | Francesco Polizzi | Sandor: you are right, I was making a confusion between the hyperplane section $\mathcal{O}_X(1)$ and the tautological sheaf $\mathcal{O}_{\mathbb{P}}(1)$, where $\mathbb{P}$ is the weighted projective space model of $X$. We have $\mathcal{O}_X(1) = \mathcal{O}_{\mathbb{P}}(2)$, so everything falls into place. | |
| Jan 7, 2015 at 23:56 | comment | added | Sándor Kovács | Francesco:This is a question of notation.I assumed that $\mathscr O_X(1)=\mathscr O_{\mathbb P^6}(1)\left|_X\right.$ as (somewhat, e.g., in Hartshorne) customary.Alternatively, it could mean the positive generator of $\mathrm{Pic}(X)$, which is the same in this case.I suppose you are thinking of $\mathscr O(1)$ as a weighted projective space, but I doubt that that was the intention of the OP. From the way the question is worded I'd expect that the interest was in the index. Other than the weighted projective case I'd never denote a non-Cartier divisor as $\mathscr O(1)$, it is too suggestive. | |
| Jan 7, 2015 at 20:17 | comment | added | Francesco Polizzi | @Sandor, it seems to me that $H= \mathcal{O}_X(2)$, so actually we have $K_X= \mathcal{O}_X(-5)$, that is $a=1$, $b=-5$. This is consistent with the description of $X$ as the weighted projective space $\mathbb{P}(1, \, 1, \, 1, \, 2)$. In fact, $\mathcal{O}_X(1)$ is not Cartier and generates $\textrm{Cl}(X)$, whereas the Cartier divisor $H=\mathcal{O}_X(2)$ generates $\textrm{Pic}(X)$. Do you agree? | |
| Jan 4, 2015 at 22:04 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 4, 2015 at 21:42 | vote | accept | CommunityBot | ||
| Jan 3, 2015 at 18:10 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 3, 2015 at 8:21 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 2, 2015 at 21:18 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 2, 2015 at 21:11 | history | answered | Sándor Kovács | CC BY-SA 3.0 |