Timeline for Deformations of smooth projective hypersurfaces and the Jacobian ring
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 18, 2012 at 7:52 | comment | added | Jack | Dear Jason-- I again thank you very much for your illuminating answers. | |
| Jul 17, 2012 at 20:46 | vote | accept | Jack | ||
| Jul 17, 2012 at 18:32 | comment | added | Jason Starr | Dear Jack -- Precisely $R_d$ parameterizes first order deformations of $X$ as an abstract projective manifold which come from first order deformations of $X$ as a closed submanifold of $\mathbb{P}^n$. This part can be seen directly: given a base space / parameter space $B$ for deformations of $X$ with local coordinates $t_1,\dots,t_d$, and given a deformation of $X$ with defining equation $F(t_1,...,t_n;x)$, then the image in $R_d$ of the generator $\partial/\partial t_i$ of $T_0 B$ is just $\partial F/\partial t_i$. | |
| Jul 17, 2012 at 18:30 | comment | added | Jack | Of course I mean a general family of smooth plane curves of degree $d$, which $d$ can be greater than 6. | |
| Jul 17, 2012 at 18:19 | comment | added | Jack | Dear Jason, Thanks alot for your complete answer. I have a question about what you said: "it is not true that the natural map $R_{d}$ to $H^{1}(X,T_{X})$ is surjective: there are plenty of deformations which are not plane curves". So if $n=2$ and we have a family of plane curves,it must be true that at least the image of the Kodaira-spencer map of the family in $H^{1}(X,T_{X})$ must be contained in $R_{d}$. Because the KS map parametrizes the infinitesimal deformations of $X$ in the family which are all plane curves and based on your argument the image of KS must be in $R_{d}$. Is this true? | |
| Jul 17, 2012 at 17:28 | history | edited | Jason Starr | CC BY-SA 3.0 |
Fixed one error
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| Jul 17, 2012 at 16:48 | history | answered | Jason Starr | CC BY-SA 3.0 |