Timeline for Infinitesimal deformations and moving cycles
Current License: CC BY-SA 3.0
7 events
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| Dec 20, 2012 at 21:49 | comment | added | Francesco Polizzi | Notice that you can have $h^0(\mathcal{O}_X(C))=1$ and $h^0(C, N_C)>0$ even if the deformations are not obstructed. For instance, take an elliptic $C$ curve on an abelian surface $A$: it does not move in a linear system, but the normal bundle of $C$ in $A$ has a $1$-dimensional family of sections: in fact, you can move $C$ into its algebraic equivalence class, using the translations in $A$. The translates of $C$ are a $1$-dimensional family of effective curves on $A$ that are NOT linearly equivalent to $C$, and that come from the $1$-dimensional family of first-order deformations! | |
| Dec 20, 2012 at 21:47 | comment | added | LMN | Ok, I see. I think you answered my question. | |
| Dec 20, 2012 at 21:37 | comment | added | Francesco Polizzi | Sorry, I do not understand well your question. An embeddeed deformation of $C$ is a flat family and $h^0$ of the structure sheaf of the fibres must remain constant. So if $C$ is reduced and irreducible, the fibres must be effective and connected, since $h^0(C, O_C) =1$. So the only resonable cycle to expect is an effective, connected cycle (maybe non-reduced and with embedded components). | |
| Dec 20, 2012 at 21:30 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Dec 20, 2012 at 21:24 | comment | added | LMN | Francesco, Thanks for your answer! At the end of the second paragraph, you mention that "...there is no effective cycle arising from a first order deformation of $C$". What if you don't require the cycles to be effective - does the same result hold? Requiring the relevant cycle to be effective seems to be too much - I would (perhaps naively) expect to find curves $C$ on a surface $X$ so that $\mathcal{O}_X(C)$ has a one dimensional space of global sections - while the normal bundle has more global sections. | |
| Dec 20, 2012 at 21:13 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Dec 20, 2012 at 21:08 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |