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Results tagged with abelian-varieties
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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.
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Books and lecture notes about Moduli spaces of Abelian varieties
Besides the chapter on Moduli in the classical textbook by Birkenhake-Lange, you might have a look at
A. Adler - S. Ramanan: Moduli of Abelian varieties, Lecture Notes in Mathematics 1644, Springer 19 …
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9
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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
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Accepted
Surfaces of general type with $q=1$
You can find plenty of examples with multiple Albanese fibres by considering surfaces isogenous to a product, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on th …
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13
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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 …
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3
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Accepted
Intersection number of divisors on abelian surfaces and its invariance under translation by ...
The answer is yes if $D_2$ is an ample curve. More generally, the following result holds:
Two ample line bundles on an abelian variety
are algebraically equivalent if and only if they differ by …
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10
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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 …
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5
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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} …
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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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3
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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 …
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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. …
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24
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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 …
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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 $$f^{\ast} \colon …
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Intuitive meaning of $k$-polarized Abelian surface?
It is well known that any polarized abelian variety is isogenous to a principally polarized one. More precisely, given any polaized abelian variety $(A, \, L)$ there exists a principally polarized abe …
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Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differe...
The two maps are naturally one the "dual" of the other. Let me give an immediate proof when $k=\mathbf{C}$.
Over the complex numbers one has $$J(C)=H^0(C, \Omega^1_C)^*/H_1(C, \mathbf{Z}),$$ hence $T …
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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 …
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