Timeline for Putting two complete varieties in a family over the projective line
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 9, 2012 at 17:07 | vote | accept | hadimath | ||
| May 9, 2012 at 16:58 | comment | added | hadimath | @ Jason: I was hoping to have an isomorphism of schemes between fibers over $0$ resp. $\infty$ and $X$ resp. $Y$, but maybe it would be enough (for what I need) to have isomorphisms from the underlying reduced schemes of these fibers to $X$ and $Y$. Is "Curves in abelian varieties over finite fields" the paper you mentioned? Thanks. | |
| May 8, 2012 at 17:38 | comment | added | Jason Starr | @hadimath: Do you want the fibers over $0$ and $\infty$ to be equal to $X$ and $Y$ as schemes, or do you want that the underlying reduced varieties of these fibers equal $X$ and $Y$? If the first, then, as pointed out, then this is extremely rare, already for curves. However, if you want the second condition, then this may be much more likely, cf. work of Bogomolov and Tschinkel over finite fields. | |
| May 8, 2012 at 16:40 | comment | added | hadimath | In this questions, variety means integral scheme of finite type (so irreducible and reduced). This implies that the dominant map from $V$ to $\mathbb{P}^1_k$ is flat. | |
| May 8, 2012 at 12:56 | comment | added | Jason Starr | Does the OP want the fibers over $0$ and $\infty$ to be reduced / smooth? If not, then I think this could be interesting, already for curves. | |
| May 8, 2012 at 8:00 | answer | added | Sándor Kovács | timeline score: 4 | |
| May 8, 2012 at 6:18 | comment | added | naf | I very much doubt that there are any reasonable general conditions. For example, if $X$ and $Y$ are curves of the same sufficiently large genus it is not always true (by results of Harris--Mumford and Eisenbud--Harris) that they can be put into a family over $\mathbb{P}^1$ but there do exist non-constant families over $\mathbb {P}^1$. | |
| May 8, 2012 at 5:48 | comment | added | Daniel Litt | Without some extra conditions, this question is trivial--do you want $V$ to be pure-dimensional? Do you want the morphism to be flat? | |
| May 8, 2012 at 5:24 | history | asked | hadimath | CC BY-SA 3.0 |