Timeline for How much information is encoded in the Jacobian-Kummer K3 surface of a curve of genus two?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 14, 2017 at 6:33 | comment | added | Sasha | One possibility to find an answer is by using global Torelli theorem for K3 and analyzing possible embeddings of the lattice $16A_1$ into the second cohomology of a K3 ($3U + 2E_8$). Probably much information in this direction can be extracted from works of Nikulin. | |
| Jun 9, 2017 at 9:43 | comment | added | Bernie | @Imeasy : Thank you. I knew that. But I don't know the behaviour of the 16 nodes, the tropes etc. with respect to the birational map of the singular Kummers $S_C$ and $S_{C'}$. So I can recover the curves, but what is their relation ship now? Must they be isomorphic? | |
| Jun 9, 2017 at 8:54 | comment | added | IMeasy | You can reconstruct the genus 2 curve by taking the double cover of the Theta divisor (which is a conic in the singular Kummer - namely the intersection of the Kummer with one of the tropes) ramified along the 6 nodes contained therein. | |
| Jun 8, 2017 at 14:58 | comment | added | Bernie | @Jason Starr : Okay. I phrased the question in terms of isomorphims and the K3 surfaces. My observation for Picard number one still remains. Can something be done for higher Picard numbers? | |
| Jun 8, 2017 at 14:56 | history | edited | Bernie | CC BY-SA 3.0 |
added 327 characters in body; edited title
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| Jun 8, 2017 at 11:26 | comment | added | Jason Starr | For K3 surfaces $S$ and $S'$, for dense open subsets $U\subset S$ and $U'\subset S'$, for an isomorphism $\phi_U:U\xrightarrow{\cong} U'$, there exists a unique isomorphism $\phi:S\xrightarrow{\cong} S'$ whose restriction to $U$ equals $\phi_U$. So instead of working in the birational category, I suggest that you work in the biregular category. | |
| Jun 8, 2017 at 10:08 | history | asked | Bernie | CC BY-SA 3.0 |