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Results tagged with abelian-varieties
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user 7460
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.
24
votes
Why polarization of abelian varieties?
Let me answer your last question "How can I think geometrically (in the lattice) about fixing a polarization?". I will follow the treatment given in [Birkenhake-Lange, Complex Abelian Varieties, Chapt …
- 66.3k
18
votes
Quotient of abelian variety by an abelian subvariety
Let us work over $\mathbb{C}$.
The inclusion $u \colon B \to A$ induces a surjection $\hat{u} \colon A^{\vee} \to B^{\vee}$.
By general facts on Abelian varieties, the kernels of $u$ and $\hat{u}$ ha …
- 66.3k
13
votes
Is every abelian variety a subvariety of a Jacobian?
Let me give an answer for $k = \mathbb{C}$.
By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian.
Now just apply Matsusaka's theorem to …
- 66.3k
11
votes
Covering of Abelian variety by product of elliptic curves
Not in general.
For instance, if $A$ is simple then this is not possible. In fact, any isogeny $$f \colon E_1 \times \cdots \times E_n \longrightarrow A$$ would give a dual isogeny $$f^{\ast} \colon …
- 66.3k
10
votes
Geometry of Albanese image
The first natural observation is the following. Set as usual $q(X):=h^{1, \,0}(X)$, so that $q(X)= \dim \mathrm{Alb}(X)$. Then the Albanese map of $X$ cannot be surjective if $\dim X < q(X)$, and it …
- 66.3k
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 …
- 66.3k
9
votes
Accepted
Why can projective varieties just have abelian group operations?
I borrow this proof from [Birkenhake-Lange, Complex Abelian Varieties, Lemma 1.1.1].
Let $X$ be a projective variety having a group structure. I assume that we are working over $\mathbb{C}$.
Consid …
- 66.3k
5
votes
A surjective morphism of abelian varieties induces an epimorphism on the torsion subgroups
Clearly the image of a torsion point by a group homomorphism is again a torsion point, so it only remains to prove the surjectivity of the restriction $f \colon A_{\mathrm{tors}} \to B_{\mathrm{tors} …
- 66.3k
5
votes
Accepted
Intuitive meaning of $k$-polarized Abelian surface?
It is well known that any polarized abelian variety is isogenous to a principally polarized one. More precisely, given any polaized abelian variety $(A, \, L)$ there exists a principally polarized abe …
- 66.3k
5
votes
About isogenies of abelian varieties
Over $\mathbb{C}$ the proof of this fact is very simple.
In fact, given any complex Abelian variety $X:= V/\Gamma$ of dimension $g$, one can find strictly positive integers $d_1, \ldots ,d_g$ and a b …
- 66.3k
4
votes
Accepted
Degree of a smooth curve in an abelian variety
Well, this is surely false for elliptic curves, i.e. when $g=1$.
What is true is that, for any $t \in \mathbb{N}$, the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. …
- 66.3k
4
votes
Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differe...
The two maps are naturally one the "dual" of the other. Let me give an immediate proof when $k=\mathbf{C}$.
Over the complex numbers one has $$J(C)=H^0(C, \Omega^1_C)^*/H_1(C, \mathbf{Z}),$$ hence $T …
- 66.3k
4
votes
Which curves can be found on Abelian varieties?
In his paper "Abelian surfaces with (1,2)-polarization" Barth proves that a smooth genus $3$ curve can be embedded into an Abelian surface if and only if it admits an elliptic involution, i.e. a map o …
- 66.3k
3
votes
Accepted
Curve through the 16 singular points of a Kummer surface
Regarding your first question, the answer is no.
Following J. Silverman's suggestion, let me provide an example of an abelian surface $A$ and a smooth curve $C$, fixed by the involution $(-1)_A$, an …
- 66.3k
3
votes
Accepted
Fiber of the Prym map in dim 2
The fibres of the extended Prym map $\overline{P} \colon \overline{\mathcal{R}}_3 \to \mathcal{A}_2$ are studied in detail in the paper
Verra, Alessandro: The Fibre of the Prym Map in Genus Three, Ma …
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