11
votes
Accepted
Which schemes are divisors of an abelian variety?
Any curve of genus greater than two, whose Jacobian $J$ is simple, will do. If it were a divisor on an abelian surface $S$, then there would be a surjection $J\to S$ with positive dimensional kernel, ...
- 2,481
9
votes
Which schemes are divisors of an abelian variety?
An obvious class of counterexamples are uniruled varieties. In fact, abelian varieties contain no rational curves.
More generally, and for the same reason, if $X$ is any algebraic variety that contain ...
- 65.1k
8
votes
Which schemes are divisors of an abelian variety?
Here's another answer using the Albanese that's of a slightly different flavor. Let $X$ be $n$-dimensional and suppose that $h^0(X,\Omega^1_X)<n$. Then any map $X\rightarrow A$ where $A$ is an ...
- 581
7
votes
Accepted
Néron model, torsion and ramification
If, for example, $A[n]$ is already contained in $A(K)$, then there will be no ramification, regardless of whether or not $A'$ is an abelian scheme (although that may well force the special fiber to be ...
- 45.8k
6
votes
Accepted
Lifting a splitting of an Abelian variety to characteristic 0
$\newcommand{\cA}{\mathcal{A}}\newcommand{\cB}{\mathcal{B}}\newcommand{\bZ}{\mathbb{Z}}$No, that does not imply that $\cA$ splits over $R$. In fact, if $\cA_1=\cA\times_R R/p$ is isogenous to a ...
- 7,027
6
votes
Accepted
hard Lefschetz isomorphism for rational Tate module
Since the question is “Can someone help... “, then “Sure, Beilinson-Bernstein-Deligne can!” seems to be a legitimate answer. Let me write a few words.
The statement you give is a variant of the ...
- 12.8k
5
votes
Accepted
extending homomorphisms of Abelian schemes
Since the OP suggested it, I am posting my comments as an answer. By the construction of Hilbert and Quot schemes, there is a relative Hom scheme, $\text{Hom}_S(\mathcal{A},\mathcal{B})$ over $S$ ...
3
votes
Accepted
Elimination of noetherian hypothesis for abelian schemes
I am just writing my comments as an answer. As nfdc23 explains, there are stronger results that require weaker hypotheses, but let me assume that $R_0$ is a finitely generated algebra contained in $R$...
2
votes
Accepted
Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?
I am posting my comments as an answer.
Let $k$ be a field. Let $$(G,m:G\times_{\text{Spec}\ k}G \to G)$$ be a locally finitely presented group scheme over $\text{Spec}\ k$. For every open $U$, ...
2
votes
Accepted
$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$
This is a special case of Pink's "canonical construction" functor, which associates various kinds of coefficient sheaves on a Shimura variety (etale $\ell$-adic sheaves, vector bundles with ...
- 36.1k
1
vote
Which schemes are divisors of an abelian variety?
I just want to point out that "adjunction+translation" tells us quite a bit:
Let $A$ be an abelian variety, say of dimension $n>1$ and let $D \subset A$ be a (let's say smooth) divisor. ...
- 778
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