Timeline for Deformation of a singular CY manifold
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2017 at 14:55 | answer | added | Basics | timeline score: 1 | |
| Feb 15, 2017 at 11:27 | comment | added | Jason Starr | Francesco is completely correct. The total space of the deformation will not be smooth. A first order deformation as hypersurfaces in $\mathbb{P}^4$ is equivalent to an element in $H^0(X\cup Y, \mathcal{O}_{\mathbb{P}^4}(\underline{X}+\underline{Y})|_{X\cup Y})$. This section vanishes on a Cartier divisor $(X\cup Y)\cap Z$ in $X\cup Y$, where $Z$ is a quintic hypersurface. The total space of the deformation is singular along $(X\cap Y\cap Z)$. This is a curve in the sextic K3 surface of degree $30$. | |
| Feb 15, 2017 at 9:48 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Feb 15, 2017 at 9:47 | comment | added | Francesco Polizzi | I think that your condition $N_{Z/X}\otimes N_{Z/Y} \cong \mathcal{O}_Z$ is equivalent to the existence of a semistable smoothing, in particular the total space of the deformation must be smooth (at least, this is the requirement in Friedman's paper Global smoothings of varieties with normal crossings). Have you checked if this is the case for the smoothing of $X \cup Y$ to a quintic threefold? | |
| Feb 15, 2017 at 6:28 | review | First posts | |||
| Feb 15, 2017 at 7:07 | |||||
| Feb 15, 2017 at 6:26 | history | asked | Lee | CC BY-SA 3.0 |