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Results tagged with abelian-varieties
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user 40297
Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
3
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A curve in an abelian surface and its image in the Kummer surface
Let $b:\hat{X}\rightarrow X$ the blowing up of the 16 2-torsion points, and $E_1,\ldots ,E_{16}$ the exceptional $(-1)$ curves on $\hat{X}$. The involution $x\mapsto -x$ lifts to an involution $\sigm …
- 38k
1
vote
Accepted
Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric ...
1) Yes. If there is another one, it differs from $L'$ by a line bundle $M$ with $M^{2}\cong \mathcal{O}_Y$. Consider the resolution $\pi :\hat{Y}\rightarrow Y$ obtained by blowing up the double points …
- 38k
3
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Picard number of abelian variety
The preprint http://arxiv.org/pdf/1310.3402.pdf may interest you (look at §3 and 4).
- 38k
3
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Derived equivalence of families of dual abelian varieties
Yes if $X$ is an abelian scheme over $B$: the Fourier-Mukai functor provides an equivalence of the derived categories. This is Theorem 1.1 in Mukai's Fourier functor and its application to the moduli …
- 38k
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Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$
I am not sure I fully understand your question, but : $A_{\mathbb{C}}$ is the quotient of a complex vector space $V$ by a lattice $\Gamma $, which is canonically isomorphic to $H_1(A_{\mathbb{C}},\mat …
- 38k
9
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Accepted
Pull-back of an irreducible ample divisor via an isogeny of abelian varieties
I think the Proposition is not true if $D$ is singular. Take a smooth curve $C$ of genus 2, and $X=JC$; embed $C$ in $X$ (say, by choosing a point of $C$). Let $\alpha$ be a point of order 2 in $X$; t …
- 38k
4
votes
Accepted
Isogeny of abelian varieties
OK, assume $\alpha _i=1$ for all $i$ (this is the only case where the question makes sense). You are given an isogeny $JC\sim A=E_1\times \ldots \times E_g$, with the $E_i$ in different isogeny class …
- 38k
6
votes
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Algebraic Hodge decomposition of CM abelian varieties
Suppose, to simplify, that $A$ is defined over $\mathbb{Q}$. Then $H^1(A_{\mathbb{C}},\mathbb{C})$ has two $\mathbb{Q}$-structures, one coming from singular cohomology, the other one from the algebrai …
- 38k
3
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Existence of divisor in the Jacobian of smooth curve of genus two whose intersection with th...
I think I understand your question now: you want your $D$ to be effective, hence irreducible since $\theta $ is ample. Now the index theorem $(D\cdot \theta )^2\geq D^2\cdot\theta ^2\ $ implies $D^2= …
- 38k
7
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Difference between stabilizer and automorphism group of subvariety of an abelian variety
They have absolutely no reason to be equal. Consider the case where $A$ is the Jacobian of a genus 2 curve $C$, and $X=C$ embedded in $A$ by $x\mapsto [x]-[p]$ for some fixed point $p\in C$. Then $X$ …
- 38k
15
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Torsion points of abelian variety as zeros of a section of a vector bundle?
The crucial case is $m=1$: if you have a vector bundle $E$ on $A$ of rank $\dim(A)$ and a section $s$ of $E$ whose zero locus is $\{0\} $, pulling back $(E,s)$ by multiplication by $m$ gives the gener …
- 38k
9
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Identifying the canonical principal polarization of a Jacobian
The answer depends very much on how you define the homomorphism $p$ associated to the polarization. My personal choice is $p(a)=\mathcal{O}(\Theta _a-\Theta )$, where $\Theta $ is a theta divisor and …
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Linear system on an abelian surface
The situation is different for abelian surfaces: if $A$ contains no elliptic curves, a linear system $|C|$ on $A$ is very ample as soon as the genus of $C$ is $\geq 8$, see this paper of Ramanan, or …
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3
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Divisors on an abelian surface
The fiber of the projection to $E'_i$, say $F_i$, is isomorphic to $E_j$ ($j\neq i$), not to $E'_j$. We have indeed $(F_1.F_2)=2$, and $F_1,F_2$ generate the group of divisors on $A$ up to numerical e …
- 38k
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can all CM types be realized by Jacobians?
It is certainly not what is expected : a conjecture of Coleman predicts that for $g\geq N$ (see below) there are only finitely many Jacobians of genus $g$ which are CM. Coleman's original conjecture …
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