11

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, contradicting the simplicity of $J$. Most curves of genus larger than two have this property; a randomly chosen example is $y^3 = x^4 - x$.

9

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 a (possibly singular) rational curve, then $X$ is not a subvariety of an abelian variety, in particular it is not a divisor there.

answered Aug 19 '20 at 18:54

Francesco Polizzi

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8

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 abelian variety factors through the Albanese, which is of dimension less than $n$, so $X$ can't be a divisor on any abelian variety. So as an example you could take ...

6

$\newcommand{\cA}{\mathcal{A}}\newcommand{\cB}{\mathcal{B}}\newcommand{\bZ}{\mathbb{Z}}$No, that does not imply that $\cA$ splits over $R$. In fact, if $\cA_1=\cA\times_R R/p$ is isogenous to a product of elliptic curves then all $\cA_n=\cA\times_R R/p^n$ are isogenous to products of elliptic curves:
Lemma. Let $S'\twoheadrightarrow S$ is a surjection of $...

6

To avoid notational confusion between translation in the derived category (of abelian fppf sheaves on the category of lfp $S$-schemes) and torsion in abelian schemes, I'll denote the $n$-torsion in $A$ as $A_n$ rather than $A[n]$, and likewise for $B$.
The $n$-torsor Kummer sequence for $B:=A^t = \mathscr{Ext}^1_S(A,\mathbf{G}_m)$ gives rise to the mapping ...

6

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 relative Hard Lefschetz theorem
for pure perverse sheaves — Théorème 5.4.10 in Beilinson-Bernstein-Deligne's paper Faisceaux pervers (Astérisque 100, 1982, Société ...

5

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 $S$ whose connected components are quasi-projective over $S$. The claim is that these components are proper over $S$. By the valuative criterion of properness, it ...

3

Always, because the dual abelian scheme/space can be defined as the connected component of the (fine) moduli space of invertible sheaves trivialized at $0$. The poincare bundle is the universal object.
PS: Defined as above it is clear from general theory that the dual abelian something is an algebraic space. It was shown I think by Raynaud that it is a ...

3

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 $R$, and let $A_0$ be a proper, flat $R_0$-scheme whose geometric fibers are reduced and whose base change $A$ to $R$ has geometrically integral fibers. By Th&...

2

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 $U$, denote by $m_U$ the restriction of $m$, $$m_U:U\times_{\text{Spec}\ k}U \to G.$$
Lemma 1. If $G$ is connected, then for every nonempty open $U$, the morphism $...

2

I recommend that you look at my joint paper with Tom Graber.
MR3114946 Pending
Graber, Tom(1-CAIT); Starr, Jason Michael(1-SUNYS)
Restriction of sections for families of abelian varieties. (English summary) A celebration of algebraic geometry, 311–327,
Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013.
14K12 (14C05)
In particular, ...

1

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. Since $\omega_A = \mathcal{O}_A$, the adjunction formula
$$
\omega_D = \omega_A(D)|_D = \mathcal{O}(D)|_D,
$$
the normal bundle of $D$. By differentiating the ...

1

Certainly it does not suffice to take the schematic closure if $S$ is nonreduced. For instance, let $S$ be $\text{Spec}\ k[x,y]/\langle x^2, xy \rangle$. Let $A$ be $E \times_{\text{Spec} k} S$, where $E$ is an elliptic curve over $k$ with specified zero point $z$. Let $p \in S$ be the closed point with maximal ideal $\langle x,y\rangle$. Let $U$ be the ...

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