Timeline for Local properties of analytic elliptic surfaces
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2010 at 18:14 | comment | added | BCnrd | Dear Lucas: Since $\pi$ is proper submersion, K-M methods apply almost verbatim. What do you & classmates with related interests not see how to carry over? The result for analytic families of curves is due to Grothendieck (try Expose 16 in 1960/1 Sem. Cartan). He might assume global projectivity, but that can be removed via theory of relative ampleness in my paper "Relative ampleness in rigid-analytic geometry" that I wrote to be applicable verbatim in C-analytic case (make the ample rel. eff. Cartier divisor via flatness and "quasi-sections" through C-M pts in a fiber, which is algebraic!). | |
| Nov 12, 2010 at 17:51 | comment | added | Lucas Culler | I meant to assume smoothness so that's not an issue. I agree that it's true for algebraic families, so assuming the last fact you mentioned I would agree that it's true for analytic families. Where would I find a proof of that fact? | |
| Nov 12, 2010 at 17:45 | comment | added | BCnrd | Dear Lucas: The question says that all fibers have genus 1 (as an aside, pf of local constancy of Euler char. relative to proper flat morphisms as in Mumford's book on abelian varieties in alg. case works verbatim in analytic case), but I didn't notice you didn't assume smoothness (since it doesn't make sense to ask #2 without smoothness). You could have degenerate fibers with many components, etc., so then K-M methods/conclusions don't apply. Will you be happier to know that loc. on the base any proper flat analytic family of curves is pullback of an analytified alg. family? | |
| Nov 12, 2010 at 17:30 | comment | added | Lucas Culler | Rather, I just don't see how the arguments in Katz-Mazur apply in this situation. | |
| Nov 12, 2010 at 17:19 | comment | added | Lucas Culler | I'm not assuming that the generic fiber has genus one. Why does this follow from my assumptions? | |
| Nov 12, 2010 at 16:52 | comment | added | BCnrd | By the way, you may enjoy the exercise (perhaps hard, but there are lots of people in Boston whom you can talk with about it) to rigorously prove that the analytified modular curves enjoy the "expected" universal property for analytic families of elliptic curves over any complex-analytic space. (This all generalizes to analytic families of higher-dimensional complex tori over complex-analytic spaces, but that needs a lot more preliminary work.) | |
| Nov 12, 2010 at 16:46 | comment | added | BCnrd | Let $D$ be any C-analytic space. By smoothness, loc. on $D$ there's a section. Hence, answer to the first is "yes": use analytic cohomology and base change (proved like in alg. case, once the basic cohomological formalism is in place, say via "Coherent Analytic Sheaves") and arguments from Ch. 1 and 2 of Katz-Mazur to prove that loc. on $D$ there's a Weierstrass model. For 2nd, answer is "yes" only loc. on the base because $L = {\rm{R}}^1 \pi_{\ast}(\mathbf{Z})$ is a rank-2 local system but may not be globally free. This is related to the univ. property of standard family over $\mathfrak{h}$. | |
| Nov 12, 2010 at 16:30 | history | edited | Lucas Culler | CC BY-SA 2.5 |
deleted 175 characters in body
|
| Nov 12, 2010 at 16:14 | history | asked | Lucas Culler | CC BY-SA 2.5 |