Timeline for Double dual of ample sheaf
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2 events
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| Jul 1, 2016 at 7:43 | comment | added | Chieh LIU | Finally, I find that this may be not true in general. If $\mathcal F$ is of rank $1$, the double dual of $\mathcal F$ is just the determinant bundle $\det(\mathcal F)$. Moreover, there is a closed subscheme $Z\subset X$ of codimension $\geq 2$ such that $\mathcal F= \mathbb P(\det(\mathcal F))\otimes \mathcal I_Z$. Since $\mathbb P(\mathcal F)$ is just the blowing-up of $\pi\colon Bl_Z X\to X$, if $Z$ is smooth, ampleness of $\mathcal F$ just means $\pi^*\det(\mathcal F)-E$ is ample, where $E$ is the exceptional divisor. In general, this doesn't imply that $\det(\mathcal F)$ is ample. | |
| Jun 29, 2016 at 10:48 | history | asked | Chieh LIU | CC BY-SA 3.0 |