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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
  • 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. ...
user avatar
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 $K$ ...
user avatar
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 $...
user avatar
  • 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), ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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/...
user avatar
  • 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 ...
user avatar
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 $n$-...
user avatar
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 ...
user avatar
  • 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_{...
user avatar
  • 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).
user avatar
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'...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
  • 116k

Only top scored, non community-wiki answers of a minimum length are eligible