Timeline for Obstruction to lifting of global sections of invertible sheaves
Current License: CC BY-SA 3.0
2 events
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| Jan 19, 2016 at 17:29 | comment | added | Hacon | Let $p\in S$ be a point on a smooth surface and $C\subset S$ a curve with multiplicity $m(m-1)>C^2$ at $p$. Let $f:X\to S$ be the blow up of $p$ and $L=\mathcal O _S(f^*C-mE)$ where $E$ is the exc. curve and $s\in H^0(L)$ corresponding to $f^{-1}_*C$ (such examples were constructed by Miranda; pg 279 Laz. book). If $X'$ and $L'$ are induced by deforming the point $p\in S$, then I think that $s$ does not lift to $H^0(L')$ (otherwise, following an argument of G. Xu, see eg proof of 2.1 of arxiv.org/pdf/0809.2160.pdf, we should have $C^2\geq m(m-1)$.....sorry I didn't check the details). | |
| Jan 19, 2016 at 14:41 | history | asked | Ron | CC BY-SA 3.0 |