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11 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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0answers
21 views

Decomposition of abelian varieties up to isogeny

Let $A_1,A_2,B_1,B_2$ be simple abelian varieties over a number field $k$. Suppose that $A_1\times A_2$ is $k$-isogenous to $B_1\times B_2$. Can we deduce that (up to reordering the factors) $A_1$ is ...
5
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1answer
77 views

Differential of a morphism of abelian varieties

I am reading the lecture notes of J.S. Milne on Abelian varieties and I got stuck at some point. Let $\alpha,\beta\colon X\rightarrow Y$ be homomorphisms of abelian varieties $X$ and $Y$. Then for ...
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0answers
28 views

$X\to \textrm{End}(O_{X,e}/m_{X,e}^r)$ is a morphism

Suppose $X^n$ is a complete group variety over algebaically closed field $k$, then the group law can be shown to be commutative. In proving this, one step is to show $X\to ...
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1answer
61 views

Isogenies of abelian varieties

Let $\pi:X\to Y$ be a finite morphism of smooth projective curves over an algebraically closed field (of characteristic zero if necessary) which are both of genus $>1$. We have two "natural" maps ...
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0answers
14 views

Endomorphism Ring of Simple Abelian Varieties

I know that if $A$ is a simple abelian variety over a number field $k$ with all endomorphisms defined over $K$ then $\mbox{End}(A_K)\otimes \mathbb{Q}$ is a division algebra with a positive ...
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0answers
38 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
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1answer
57 views

Examples of varieties with torsion in their integral Hodge structure

I am not so used to thinking about integral Hodge structures, so this question might be completely trivial. What are easy and interesting examples of smooth projective connected varieties $X$ with ...
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1answer
30 views

Is finiteness of rational points preserved by duality?

Sorry if this is obvious. I don't know much about Abelian varieties. Let $A/k$ be an abelian variety. Let's say $k$ has characteristic zero. Let $\widehat{A}$ be the dual abelian variety. Suppose ...
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0answers
15 views

is it possible to express the moduli of ppav's using torelli loci?

This is a probably vague question from an outsider: It is well known that it is possible to embed "the" moduli space of curves $M_g$ with fixed genus g into the moduli space of principally polarized ...
0
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1answer
28 views

Dual isogenies of complex tori in Birkenhake-Lange

Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$. Basically, they show ...
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0answers
30 views

The characteristic polynomial of Frobenius of an RM curve

Let $C$ be a genus two curve over $\mathbb{Q}$. We can reduce $C$ modulo a prime $p$ to obtain a curve $\bar{C}$ over $\mathbb{F}_p$. By counting points of $\bar{C}$ over $\mathbb{F}_p$ and ...
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1answer
40 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
3
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1answer
179 views

Picard group and jacobian of a singular curve

I found somewhere written that the jacobian of a reducible curve is the product of the jacobians, which seems quite reasonable to me. Where can I find some proof of it and a description of the Picard ...
3
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1answer
90 views

Subtori of a complex torus

I am beginning to read Birkenhake-Lange, Complex Abelian Varieties, where they define a complex torus as being a quotient $X=\mathbb{C}^g/\Lambda$, where $\Lambda$ is a lattice in $\mathbb{C}^g$. ...
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0answers
111 views

Deformation polarized abelian variety

In "$C$ is not algebraically equivalent to $C^{-}$ in its Jacobian" by Ceresa and then in "On the periods of certain rational integrals, II" by Griffiths (which is quoted as a reference in the first ...
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1answer
141 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
4
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1answer
57 views

What is this cycle on the Jacobian of a curve?

Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree ...
5
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1answer
83 views

Etale subgroup-scheme

Let $A$ and $B$ be two abelian varieties over a field $k$ and let $l$ be a prime number not dividing the caracteristic of $k$. Let $\phi : A \to B \in Hom(A,B)$ be such that $\phi$ is zero on ...
1
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1answer
69 views

Abelian Varieties over C vs. Abelian Varieties over C_p

Suppose $A$ is a complex abelian variety. Then $A$ is a complex torus $\mathbb C^g/\Lambda$ where $\Lambda$ is a lattice. On the other hand abelian varieties over $\mathbb C_p$ can have good ...
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1answer
42 views

Group schemes appearing in the $p$-torsion of an abelian scheme

This question is motivated by this one. In that question, there is stated without reference the fact that for an abelian variety over a field of characteristic $p$ (which I assume means the base field ...
2
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1answer
71 views

calculate the group of all biholomorphic group automorphisms of complex tori

My backgrand is complex geometry,but when I confront complex tori,I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group. Let ...
4
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2answers
124 views

Tate module of product of abelian varieties

Is there a way to relate the Tate module $T_{l}(A \times B)$ of the product of two abelian varieties $A$ and $B$ over a field $k$ (where $l \neq \text{char}(k)$), to the Tate modules $T_{l}(A)$ and ...
2
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0answers
55 views

Trivial divisor on elliptic curve

Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in ...
3
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2answers
109 views

Pullback of a vector bundle on Abelian variety via $(-1)$

Let $A$ be an abelian variety over some field $k$ and $(-1) : A \to A$ is the inverse map of $A$ as an algebraic group. If $V$ is a vector bundle over $A$ what is $(-1)^* V$? In other words, is there ...
2
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0answers
45 views

Specifying the real multiplication on a curve

Suppose we have a curve $C$ of genus two over an algebraically closed field of characteristic zero with a given Weierstrass point $p_0$, furthermore we assume this curve to have real multiplication by ...
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1answer
549 views

What is the Albanese map good for?

I am reading a textbook on complex manifolds and come across Albanese map. For a compact Kaehler manifold $X$, $$ T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z}) $$ is a complex torus, called the Albanese ...
6
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1answer
121 views

Is $M_g$ NEVER proper? And why does $T_g$ contain products?

Work over an algebraically closed field ($\mathbb C$, if you prefer) and fix $g\geq 2$. By $M_g$ I mean, of course, the moduli space of smooth projective curves of genus $g$. I know it is in general ...
5
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1answer
195 views

Translation of a Paper of Tate

I'm just wondering if anyone knows if the paper "Classes d'isogenie des varietes abeliennes sur un corps fini" by John Tate has been translated into English. My French is not that good and I found it ...
3
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1answer
105 views

“Néron-Ogg-Shafarevich criterion” in positive characteristic

The Néron-Ogg-Shafarevich theorem usually seems to be cited to say that an abelian variety over a finite extension $K/{Q}_p$ has good reduction at $\ell \neq p$ if and only if the associated ...
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2answers
356 views

Divisors in an abelian surface

How to compute the Néron-Severi group of the abelian surface $Y = \mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i]$. More generally, are there any result that compute the Néron-Severi group of ...
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1answer
168 views

Why does Mumford want to avoid “reduction to Jacobians”?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
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0answers
92 views

Triviality of the tangent space of an abelian variety

The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all ...
3
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1answer
104 views

Does the Euler characteristic increase if I add an effective divisor

Let $D$ be an effective divisor on a smooth projective connected complex algebraic variety $X$. Suppose that $D\leq E$. Is it true that $$\chi(X,\mathcal{O}_X(D)) \leq \chi(X,\mathcal{O}_X(E))?$$
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1answer
139 views

Linear system of divisors on complete variety

I am currently reading Mumford's abelian varieties and Milne's notes on them and I have a problem understanding the proof that they are projective. Both of them use that a complete linear system of ...
2
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0answers
40 views

Theta group representation

Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ ...
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0answers
53 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
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0answers
32 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
4
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0answers
138 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
2
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1answer
154 views

Abelian subvarieties of a principally polarized abelian variety are principally polarized

Let $A$ be a principally polarized abelian variety. Let $X\subset A$ be an abelian subvariety. Is $X$ also principally polarized? Here's what I think should be a proof. Is it correct? We may and do ...
3
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1answer
81 views

(Non)-Isomorphic Jacobians

I've started reading Hindry-Silverman's Diophantine Geometry and I got a little ahead of myself. Non-isomorphic elliptic curves have non-isomorphic Jacobians, that is because elliptic curves are ...
4
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2answers
142 views

Subvariety of Product of Elliptic Curves

This is almost certainly known (and maybe written down somewhere?). Is there an example of two elliptic curves $C, E/k$ that are not isomorphic, yet there is an embedding $C\hookrightarrow E\times E$ ...
4
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1answer
198 views

Jacobian of a curve

Let $C$ be a curve and $J$ be its Jacobian. What is the relation between $H^1(C,\mathcal{O}_C)$ and $H^1(J,\mathcal{O}_J)$ ? Can someone point me to an easy reference for this subject?
4
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1answer
271 views

Canonical divisor of an abelian variety

Let $A$ be an abelian variety over a field $k$ and let $K_A$ be its canonical divisor. Then I'm almost certain that $K_A$ is trivial, but I can't seem to prove it, nor find a counter example, nor ...
3
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0answers
117 views

Some complex torus is not an abelian variety

Let $e_1,...,e_4\in\mathbb{C}^2$ whose coordinates are all algebraically independent. Let $\Lambda$ be the lattice spanned by these vectors. Why is $\mathbb{C}^2/\Lambda$ not an abelian variety?
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0answers
112 views

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
2
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0answers
120 views

Notation for a canonical quotient of an abelian variety in positive characteristic

This may be a somewhat silly question, but there it goes. Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian variety of dimension $g\geq1$. ...