Timeline for Poincaré bundle and Weil pairing for Abelian schemes
Current License: CC BY-SA 3.0
8 events
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| Mar 7, 2014 at 14:41 | comment | added | user76758 | OK, this is what I think is usually called the "rigidified" Poincar\'e bundle (typo: you missed an inversion on the 2nd tensor factor). But encoding such rigdification is part of the very content of building a Poincar\'e sheaf on the entire Picard scheme (or algebraic space), so I remain puzzled as to where the point of confusion is arising for Pic$^0$ versus Pic in terms of Poincar\'e bundles (i.e., if you are happy for Pic then why not for Pic$^0$?). | |
| Mar 7, 2014 at 14:01 | comment | added | user19475 | The normalised Poincaré bundle is $\mathscr{P} \otimes f_{A^\vee}^*g_{A^\vee}^*\mathscr{P}$ with $f: A \to X$ and $g$ the zero section. | |
| Mar 7, 2014 at 13:55 | comment | added | user76758 | If you have a given abelian scheme $B$ with line bundle $L$ on $A \times B$ equipped with trivialization $i$ of its pullback to $A \times \{0\}$ then to check if the resulting map $B \rightarrow A^{\vee}$ is an isomorphism (thereby giving the universal property to $(B, L, i)$ it suffices to check on geometric fibers, where various results in Mumford's book are applicable. I don't know what "modified Poincar\'e bundle" means (FGA Explained not nearby at the moment), but would that address whatever is concerning you? | |
| Mar 7, 2014 at 13:47 | comment | added | user19475 | The question is if the universal property still holds (for the modified Poincaré bundle as in [FGA explained, Kleiman, The Picard scheme], p. 289, Remark 9.5.24. | |
| Mar 7, 2014 at 13:44 | comment | added | user76758 | The functor ${\rm{Pic}}^0_{A/S}$ is a subfunctor of ${\rm{Pic}}_{A/S}$ (defined by a condition on geometric fibers), so what is the meaning of the question in the "Edit" that isn't a tautology (via pullback)? | |
| Mar 7, 2014 at 13:25 | history | edited | user19475 | CC BY-SA 3.0 |
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| Mar 7, 2014 at 13:17 | history | edited | user19475 | CC BY-SA 3.0 |
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| Mar 7, 2014 at 13:04 | history | answered | user19475 | CC BY-SA 3.0 |