# Tagged Questions

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. (Def: http://en.m.wikipedia.org/wiki/Abelian_variety)

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### Pullback of global sections?

Let $X$, $Y$ be abelian varieties and let $f:X\to Y$ be a morphism. They told me that we can define the pullback $f^*s$ of a global section $s\in\Gamma (L)$ where $L$ is an ample line bundle on $Y$, ...
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### A question on very ample line bundle on Abelian Varities

I have a problem with some consideration that Mumford does about very ample line bundles in the prove of Riemann-Roch theorem. Namely, he says that if we consider a very ample line bundle $L=O(D)$ on ...
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### Is any abelian subvariety of an abelian variety closed?

I am wondering because Milne here in Proposition 10.1, page 42, takes any abelian subvariety $B$ of an abelian variety $A$, with $0\neq B\neq A$, then he takes an ample line bundle $\mathcal{L}$ on $A$...
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### Global sections and divisors

I'm trying to understand the proof of the Theorem at page 163 from Mumford, Abelian Varieties, and I have a question about one step. This is the situation: $X$ is an abelian variety (hence there's ...
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### Definition of pullback of a Weil divisor on an abelian variety?

We are on an abelian variety, so Cartier divisors, Line bundles and Weil divisors are all equivalent. I would like to see the pullback of a Weil divisor. Is it true that, if $D=\sum n_i E_i$, then the ...
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### Connected component of $0$: why is it an abelian variety?

With "Abelian variety" I mean a integral scheme $X$, proper over an algebraically closed field (complete variety) with a group law $m: X\times X \rightarrow X$ such that $m$ and the inverse map are ...
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### Degree of étale morphism $\pi:X\rightarrow X/F$

Let $X$ be an abelian variety and let $F\subset X$ be a finite group. Then we get an étale morphism $\pi:X\rightarrow X/F$ (see construction of $X/F$ and $\pi$ in Mumford, Abelian varieties from page ...
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### Why is it true that $Supp (T^*_xD)=SuppD-x$?

Let $X$ be an abelian variety, let $T_x: X\rightarrow X$, $z\mapsto z+x$ be the translation morphism, let $D=\sum n_iY_i$ be a Weil divisor. As I said here: Pullback of a Weil divisor? I am trying to ...
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### Pullback of a Weil divisor?

How to define the pullback of a Weil divisor $D=\sum n_iY_i$? I'm particulary interested in $T^*_xD$ where $x\in X$, $X$ is an abelian variety and $T_x: X\rightarrow X$, $z\mapsto z+x$ is the ...
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### Certain map of modules is iso [Mumford Abelian Varieties]

I have trouble showing the following in the proof of Prop. 2 in Abelian Varieties (pg.70 my edition, Chapter about quotients by finite groups): Suppose you have a Noetherian ring $B=A^G$ as ...
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### How to apply Riemann-Roch Theorem and Kodaira Vanishing Theorem to an ample line bundle?

Let $X$ be an abelian variety. In "Mumford, Abelian Varieties" the Riemann-Roch Theorem has the following form: For all line bundles $\mathcal{L}$ on $X$, if $\mathcal{L}\cong\mathcal{O}_X(D)$, ...
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### $A$ abelian variety. Is the multiplication by $n_A$ surjective?

According with Mumford the answer is yes, but there are some obscure points in the proof. We know that there exists a very ample line bundle on $A$ since every abelian variety is projective. Hence, ...
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### Why is the index $i(\mathcal{L})$ of an ample line bundle on an abelian variety equal to $0$?

I've seen that here https://www.math.uchicago.edu/~ngo/Shimura.pdf there's a theorem called Mumford's Vanishing Theorem (Theorem 2.2.2) which says: Let $\mathcal{L}$ be a line bundle on $X$ (...
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### Dimension of an abelian subvariety in the proof of the Poincaré's Reducibility Theorem

I am trying to understand the proof of the Poincaré's Reducibility Theorem, that I'm reading from the book "Abelian varieties" of J.S. Milne (see Proposition 10.1) and from the book "Abelian varieties"...
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### Where does the terminology “fake/false elliptic curve” come from?

In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an ...
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### Why is $\mathcal{O}_F \otimes \mathbb{Z}_p$ a Dedekind ring?

In the paper Compactifications de l'espace de modules de Hilbert-Blumenthal (Compositio. '78), M. Rapoport relates some assertions from a letter of Deligne to Serre, dated 1972. At the very beginning, ...
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### Ribbons of abelian varieties

Let $X$ be a scheme over $\mathbb C$, such that $A=X_{\textrm{red}}$ is an abelian variety. Suppose the tangent spaces to $X$ are all of dimension $\dim X+1$. Then $X$ is non-reduced everywhere, but I ...
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### Explicit models for abelian surfaces

Is there a nice descriprion of birational models for abelian surfaces of given polarization degree?
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### A curve and its translate are linearly equivalent in a complex torus?

Let $X=\frac{V}{\Lambda}$ be a complex torus of dimension $2$. Suppose $C$ is a curve in $X$. Consider $C+a=C_1$, the curve which is a translate of $C$ by $a\in X$. Are $C$ and $C_1$ linearly ...
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### Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
Let $A$ be an abelian variety (over an algebraically closed field $k$) and $X\subset A$ a nonsingular subvariety with ideal sheaf $\mathscr I_X\subset \mathscr O_A$. Let $\tau_a:A\to A$ be the ...