Timeline for Deformations of the moduli space of ppav's
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2016 at 19:01 | comment | added | roy smith | oops, I wasn't paying close attention. | |
| Feb 16, 2016 at 6:22 | comment | added | Christian | @roysmith thank you for your comment. As far as I can tell, Beauville writes about the tangent space to $X$, and not the tangent space at $X$ in the moduli of log-canonically polarized varieties. That is, I am interested in the deformations of $X$ itself, not of its objects. | |
| Feb 16, 2016 at 5:23 | comment | added | roy smith | try page 380 of Beauville's doctor of science paper: math1.unice.fr/~beauvill/pubs/prym.pdf | |
| Feb 15, 2016 at 16:35 | comment | added | Will Sawin | I believe $T_X$ is the same as $\operatorname{Sym}^2 $ of the relative tangent bundle of the universal family. | |
| Feb 15, 2016 at 16:18 | history | edited | Christian | CC BY-SA 3.0 |
Jason Starr explained what the deformation space to $X$ is in the comments below.
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| Feb 15, 2016 at 16:14 | comment | added | Christian | @JasonStarr Thank you again. It seems that my intuition was completely wrong. | |
| Feb 15, 2016 at 15:22 | comment | added | Jason Starr | Christian: I would advise you to write down one example. For instance, if $\overline{V}$ is a smooth, projective curve of genus $g>1$, if $D$ is an effective Cartier divisor of degree $1$ on $\overline{V}$, and if $V$ is the open complement of $D$, then $\text{Ext}^1_{\mathcal{O}_V}(\Omega_{V/k},\mathcal{O}_V)$ is the zero vector space, while $H^1(\overline{V},T_{\overline{V}/k}(\text{log}\ D))$ is nonzero. | |
| Feb 15, 2016 at 15:16 | comment | added | Christian | @JasonStarr Thank you for your comment. If $V$ is log-canonically polarized with Hilbert polynomial $h$, will the dimension of $Ext^1(\Omega, \mathcal O)$ be equal to the dimension of the tangent space to the object $[V]$ in the stack of log-canonically polarized varieties with Hilbert polynomial $h$? Or is this $Ext^1$ possibly bigger than this tangent space? | |
| Feb 15, 2016 at 15:05 | comment | added | Jason Starr | For any finite type scheme $V$ over a field $k$, whether or not it is smooth, projective, etc., the vector space of first order deformations of $V$ as a $k$-scheme equals $\text{Ext}^1_{\mathcal{O}_V}(\Omega_{V/k},\mathcal{O}_V)$. | |
| Feb 15, 2016 at 14:41 | review | First posts | |||
| Feb 15, 2016 at 14:49 | |||||
| Feb 15, 2016 at 14:41 | history | asked | Christian | CC BY-SA 3.0 |