Timeline for curve over higher dimensional basis with 0-dimensional locus of bad reduction
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 7, 2015 at 17:35 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Nov 7, 2015 at 14:11 | answer | added | Laurent Moret-Bailly | timeline score: 8 | |
| Nov 6, 2015 at 17:15 | comment | added | Jason Starr | Hmm ... there is something fishy about my counterexample above. Let me think this over and post something better. | |
| Nov 6, 2015 at 14:08 | comment | added | Jason Starr | Matt DeLand, Complete families of linearly non-degenerate rational curves, arxiv.org/abs/0710.5713v1. M. Chang and Z. Ran. Closed Families of Smooth Space Curves. Duke Mathematical Journal 52(1985), no. 3, 707-713. Steven Diaz. A bound on the dimensions of complete subvarieties of ${\cal M}_{g}$. Duke Math. J. 51 (1984), no. 2, 405–408. 14H10 (14H30) | |
| Nov 6, 2015 at 13:51 | comment | added | user19475 | Can you give me the precise reference for Chang-Ran, DeLand and Diaz? | |
| Nov 6, 2015 at 13:38 | comment | added | Jason Starr | For genus $0$ curves of degree $d$ in $\mathbb{P}^n$, I believe the sharp results are known by Chang-Ran and DeLand. In terms of your $S$, there are families with $\text{dim}(S) = n$. For higher genus curves, I believe this is open. I realize that you are asking a local question, but for the associated global question, the best known result is Diaz's theorem. | |
| Nov 6, 2015 at 13:06 | comment | added | user19475 | Thank you very much! Are there more examples? | |
| Nov 6, 2015 at 12:53 | comment | added | Jason Starr | Yes. Start with the (everywhere smooth) family of lines in $\mathbb{P}^3$. Now consider the map $\mathbb{P}^3\to \mathbb{P}^3$ by $[x,y,z,w]\mapsto [x^3,y^3,z^3,w^3]$. Now consider the family of images under this map of the lines. Families such as these were studied by Matt DeLand in connection to extensions of Bend-and-Break (ala Chang-Ran). | |
| Nov 6, 2015 at 12:08 | history | asked | user19475 | CC BY-SA 3.0 |