17
votes
Is every abelian variety a subvariety of a Jacobian?
Embed the dual abelian variety into projective space. Take a smooth hyperplane section and interate until it's one-dimensional, obtaining a smooth curve $C$. By Lefschetz $C$ is irreducible, and the ...
- 116k
14
votes
Accepted
An explicit equation for $X_1(13)$ and a computation using MAGMA
To get a model with good reduction at $2$, take $y = 2Y + x^3 + x^2 + 1$,
subtract $(x^3+x^2+1)^2$ from both sides, and divide by $4$ to get
$$ Y^2 + (x^3+x^2+1) \, Y = -x^5-x^3+x^2-x. $$
(A similar ...
- 71.6k
13
votes
Good lecture notes/books on Jacobian of hyperelliptic curve
Over $\mathbb C$, there's also Mumford's beautiful monograph Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., 1975. But a monograph covering all of the topics in your ...
- 42.3k
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 ...
13
votes
Accepted
Embedding of a derived category into another derived category
Any fully faithful functor from $D^b(\mathcal{A})$ has adjoints (because $D^b(\mathcal{A})$ is a smooth and proper category), so its image is an admissible subcategory. A recent result from Dmitrii ...
- 31.8k
11
votes
Accepted
functions with orthogonal Jacobian
Such maps are conformal. A theorem of Liouville says that if $n\geq 3$,
the only conformal maps (defined in some region in $R^n$) are Mobius. A Mobius map is a composition of inversions
in spheres. ...
9
votes
Non-linear rational recurrence related to the Jacobian of hyperelliptic curve
NEW (Oct 27, 2017)
I can now show that the recurrence given in the Question is correct.
In fact, I can deduce a simpler and shorter recurrence relation.
For this, we consider the Kummer Surface $K$ ...
- 10.6k
9
votes
Accepted
If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?
No. Choose a field $k$ and $D=k[u^2,u^3,v^2,v^3,uv]\subset k[u,v]$, so $D$ is a noetherian domain. In $D[x,y]$, choose $f=(ux+vy)^2$ and $g=(ux+vy)^3$; they are clearly algebraically dependent (but $...
- 51.7k
9
votes
Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?
This is (exactly as stated in your question) for example in Proposition 10.1 of
Oda, Tadao; Seshadri, C. S. Compactifications of the generalized Jacobian variety. Trans. Amer. Math. Soc. 253 (1979), ...
- 3,072
9
votes
Accepted
2-Torsion in Jacobians of Curves Over Finite Fields
I think $y^2=x^9-x$ over $\mathbb{F}_3$ has $J_C(\mathbb{F}_3)$ isomorphic to $(\mathbb{Z}/2)^6$ but please check.
The $2$-torsion in $J_C$ over the algebraic closure is $(\mathbb{Z}/2)^{2g}$ (or ...
- 29.5k
9
votes
Accepted
Torsion in the jacobian of a super elliptic curve
As mentioned by the previous answers, this cannot be true for $n \ge 3$ by size considerations. When you identify the points $(x_i, 0)$ of $C$ inside its jacobian $J$, you are implicitly using some ...
- 106
9
votes
Accepted
Surjectivity of the Abel-Prym map
First of all, note that your definition is not correct: when $d$ is odd, the image of your map does not land in the Prym variety -- you have to add a constant term. When this is done, the answer is ...
- 34.5k
9
votes
Accepted
What is the involution on the moduli space of genus 3 curves induced by the Torelli map
The Torelli morphism being a double cover is purely a stacky phenomenon. It is not visible on coarse moduli spaces.
The involution you ask about is supposed to act on the fibers of the Torelli ...
- 37.4k
8
votes
functions with orthogonal Jacobian
$\def\RR{\mathbb{R}}$A brute force approach shows that there are no other $C^2$ solutions. Let $F: \RR^n \to \RR^n$ have orthogonal Jacobian everywhere. We will show that the Hessian of $F$ vanishes ...
- 139k
8
votes
Accepted
Do all simple factors of jacobians of curves come from correspondences?
This argument is mostly contained in t3suji's comments, but with some of the proofs somewhat expanded.
As a convention (consistent with that of the theory of Chow motives), all actions of ...
- 20.3k
8
votes
Accepted
Neron model: can number of components decrease after based change?
Yes, there is such a result for Neron models of abelian varieties with semiabelian (aka semistable) reduction: "the number of components of the special fiber cannot decrease after base change". This ...
- 4,739
8
votes
Determinant of Jacobian and directional derivatives
Any matrix $A$ can be written as $B.U$ where $B=\sqrt{AA^\top}$ is posititive semidefinite and $U$ is orthogonal (polar decomposition). Thus $|\det(A)| =|\det(B)|.|\det(U)|$ is the product of the ...
- 24.4k
7
votes
About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve
The equation for $\# J_C(\mathbb F_p)$ that you quote contains a typo: they must have meant that $\# J_C(\mathbb F_p) = \frac 1 2 \# C(\mathbb F_{p^2}) + \frac 1 2 \# C(\mathbb F_p)^2 - p$, which is ...
- 37.4k
7
votes
Accepted
A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero
The reverse implication is true in a considerably more general setting (Burchnall-Chaundy theory). Namely, for any pair $(U,V)$ of commuting meromorphic coefficient differential operators in one ...
- 14k
7
votes
Accepted
Is every abelian variety a subvariety of a Jacobian?
You can find a detailed proof here (theorem 1.2) in the case of principally polarized abelian varieties. One reduces to this case using the Zarhin's trick.
The assumption of $k$ being infinite should ...
- 637
7
votes
Accepted
Naive question on the Jacobian of a curve
It is possible for the Jacobian's of non-isomorphic curves to be isomorphic as abelian varieties, but obviously, not as principally polarized abelian varieties. This paper https://arxiv.org/pdf/math/...
- 954
7
votes
Accepted
$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?
$\def\ZZ{\mathbb{Z}}\def\QQ{\mathbb{Q}}$No. As explained in this question, in $\mathbb{Q}[x,y]$, the condition $\operatorname{Jac}(f,g)=0$ implies that there exists an $h \in \mathbb{Q}[x,y]$ such ...
- 139k
7
votes
Accepted
Relation between the cohomology group of a curve and the cohomology group of its jacobian
$\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-...
- 139k
6
votes
Accepted
What is our current knowledge on the structure of J_0(N)(Q) and J_1(N)(Q)
For torsion subgroups, one can consider the "rational cuspidal subgroup", which is the subgroup of degree zero divisors generated by $\mathbb{Q}$-rational divisors coming from cusps. (The cusps ...
- 17.8k
6
votes
Accepted
Resolution of the ideal of the Abel-Jacobi image of a curve?
This is not true, as soon as $g\geq 3$. Taking Chern classes this would imply that $c_{g-1}(\mathcal{O}_{a(C)})$ is an integral multiple of $\ \Theta ^{g-1}$ in $\ H^{2g-2}(JC,\mathbb{Z})$. But $c_{...
- 34.5k
6
votes
Accepted
Jacobians of genus 2 curves isogenous to a square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$
Such curves are constructed in my paper "Familles de courbes et de variétés abéliennes sur $\mathbb{P}^1$, II", Astérisque vol. 86 (1981).
- 18.1k
6
votes
Accepted
Abel-Jacobi map for Mumford curves analytically
This is done in Manin and Drinfeld's "Periods of p-adic Schottky groups." Journal für die reine und angewandte Mathematik 0262_0263 (1973): 239-247. You will also find it in Gerritzen and van der Put'...
- 3,697
6
votes
Accepted
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
There is a duality between degree 2 coverings and two-torsion points on the Jacobian — i.e. both form elementary abelian 2-groups, and these groups are naturally dual.
This is the Artin–Milne Poincaré ...
- 116k
5
votes
Accepted
Perfectness of the Jacobian of a curve
No in general, for instance if $K$ is a number field : assuming that $C$ has a rational point, the group $\mathrm{Pic}(C)$ is naturally isomorphic to $JC(K)$, the group of rational points of the ...
- 34.5k
5
votes
Accepted
Distribution of Mordell–Weil ranks of higher genus curves
General Katz-Sarnak heuristics suggest that the analogue of the minimalist conjecture should still be true. Let me sketch the reason why from two perspectives - the function field model, where we can ...
- 116k
Only top scored, non community-wiki answers of a minimum length are eligible