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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 $y^3 = x^4 - x$.

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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.

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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 ...

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$\newcommand{\cA}{\mathcal{A}}\newcommand{\cB}{\mathcal{B}}\newcommand{\bZ}{\mathbb{Z}}$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$...

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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, ...

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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 ...

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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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