# Questions tagged [jacobians]

The jacobians tag has no usage guidance.

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### Generic properties of Jacobians of smooth functions

Let $f = (f_1, \dotso, f_n):\mathbb{R}^n \to \mathbb{R}^n$ be a smooth map and let $J$ be its Jacobian (determinant of the matrix with $ij$-th entry $\partial_i f_j$). We introduce the zero sets of $J$...

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### $A,B,w \in \frac{\mathbb{Q}[t]}{(t^2-1)}[x,y]$: $\operatorname{Jac}(A,B)=1$, $\operatorname{Jac}(A,w)=0$, $w \notin \frac{\mathbb{Q}[t]}{(t^2-1)}[A]$

Let $R=\frac{\mathbb{Q}[t]}{(t^2-1)}$; trivially, $R$ is not an integral domain, since $(\overline{t-1})(\overline{t+1})=\overline{t^2-1}=\overline{0}$.
Is it possible to find $A,B,w \in R[x,y]$ ...

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### Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$,
and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree).
I wish to prove or find a counterexample to the following claim:
If ...

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200 views

### $f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...

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### 2-Torsion in Jacobians of Curves Over Finite Fields

Let $C$ be a (smooth, projective) curve over a finite field $\mathbb{F}_q$, and let $J_C(\mathbb{F}_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves ...

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229 views

### Confusion in known result about Moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2

Theorem: Let $X$ be a complete, non-singular algebraic curve of genus
$2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector
bundles of rank $2$ and degree $\Theta$. The ...

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### When does the Jacobian of a smooth curve contains an unique principal polarization

Let $X$ be a smooth, projective curve of genus at least $4$ and $X$ non-hyperelliptic. I am looking for additional conditions on $X$ such that the Jacobian $J(X)$ of $X$ contains an unique principal ...

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### Naive question on the Jacobian of a curve

Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...

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236 views

### Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?

Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks ...

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### Is every abelian variety a subvariety of a Jacobian?

Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?
In these notes by William Stein this is stated without proof in remark 1....

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### Some curves on the Jacobian of a genus $2$ curve and their image under certain maps (char $p$)

I hope this question belongs here. The situation in this question is quite particular and specific.
I am trying to weak some of theory to measure the degree of some function on the Jacobian of a ...

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### An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...

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182 views

### Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping

Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the
Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$.
It is well ...

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### Factors of the Jacobian of modular curves

Let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$ for p an odd prime. We know that $J_1(p)$ is isogenous to a direct sum of abelian varieties $\oplus_{f}A_f$ where the sum runs over Hecke/...

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### How to compute explicit equations for the Jacobian of a variety over a field [duplicate]

Suppose we start with a projective curve $X$ over a field $K$, given as a closed subvariety of $\mathbb P^n_K$ by some explicit list of equations. I would like to find an explicit representation of ...

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### On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?

Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...

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237 views

### A map from a symmetric product of a curve to its Jacobian

Let $C$ be a smooth projective curve over an algebraically closed field $k$, of genus $g$.
It is well known that, after fixing a point $p_0$, the map $C^{(n)}\to J$ sending $\{a_1,\dots,a_n\}$ to $[...

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### Determinant of Jacobian and directional derivatives

I have a function $f: \Re^2 \to \Re^2$ and would like to understand why
$$|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$$
that is, why the determinant of the Jacobian of $f$ at $...

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### Picard rank of Jacobian

Let $C$ be a curve over field $k$, let $J$ be its Jacobian.
(1)Suppose $C$ has no nontrivial automorphism, $k$ is algebraically closed, is it true that $\mathrm{NS}(J)\cong\mathbb{Z}\cdot\theta$? ...

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### Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points

I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.
Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D_1,...

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### If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?

This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$,
...

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### A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero

Let $f,g \in \mathbb{C}[x,y]$.
There is a well-known result, that can be found for example
here, pages 19-20, that says the following:
$f,g$ are algebraically dependent over $\mathbb{C}$ if and ...

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### Scalar multiplication via the Kummer surface of a genus $2$ curve by $\sqrt{5}$

I hope this is a good question.
Recently I worked with genus two curves $H$ that have multiplication by $[\zeta_5]\in \text{Aut}(H)$, that is, multiplication by $e^{2\pi i/5}$. This automorphism is ...

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### Connected component of Picard group of a genus $2$ curve and its Jacobian over an imperfect field

Let $H/k$ be a genus $2$ curve. Consider the function field of $H$ given by $k(H)$. Take the base extension of $H$ to $k(H)$, namely $\hat H:=H\otimes_k k(H)$. Consider the Jacobian $J$ of the curve $...

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### Non-linear rational recurrence related to the Jacobian of hyperelliptic curve

EDIT The initial revision had typo, hope it is fixed.
Let $C$ be the hyperelliptic curve $y^2=x^5+x+5^2$ and $J$ its Jacobian.
Let $P=J(0,5)$ and define $a(n)$ to be the constant coefficient of
the ...

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### Is there a finite number of supersingular genus 2 curves?

Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a ...

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### Explicit endomorphisms of Jacobians of genus $2$ and the Theta divisor

I hope this question is good to be here.
Let $J$ be the Jacobian of a hyperelliptic curve $H$ of genus $2$ given by $y^2 = x^5 + h$.
I was calculating an explicit formula in Mumford coordinates of $[...

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### 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians

I would like to know if there is something I can read to compute the following:
Let $H$ be a hyperelliptic curve of genus $2$ given by $y^2=x^5 + 10$ and let $J$ be its Jacobian.
How can I prove ...

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### How to describe the subspace of invariants under the Rosati involution?

Consider the Jacobian $J_C$ of hyperelliptic curve
$$C\!: y^2 = x^5 + a$$
over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \...

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### The existence of the Drinfeld shtuka function

I want to understand the existence of the Drinfeld shtuka function but unfortunately I know very little in algebraic geometry.
I am reading Shtukas and Jacobi sums from D. Thakur and I am stucked at ...

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### Abel-Jacobi map for Mumford curves analytically

Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL_2(K)$, and let $X_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid ...

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### Boundary of the image of a compact manifold in the complex plane

The Question
Consider the trace of an $n \times n$ unitary matrix with determinant 1
\begin{align}
f: SU(n) &\rightarrow \mathbb{C}\\
U \mapsto \text{tr}\, U &= \sum\limits_{i=1}^{n-1} z_i + ...

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### Clarke generalized Jacobian of an inverse function

For a Lipschitz function $f: X \rightarrow X$, Clarke's generalized Jacobian at $x$ is defined as the convex hull of the following set:
$$\delta f (x) = \text{convex hull} \left \{\lim_{x_i \...

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### Self intersection of theta divisor

I hope my question is not too basic here.
I have a very specific setting and I am trying to not use "big theorems" to prove a well known fact algebraically. I hope someone can give me a hint.
Let $J/...

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### How is the Jacobian or Sandpile group of a graph computed?

From what I understand, given a graph, the Jacobian group and the Sandpile group refer to the same object. Until now, I have been computing this group in the way detailed in Chapter 1 of this ...

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### Isogeny from kernel in higher dimensional abelian varieties

Is there any kind of generalization of Vélu formulae for Jacobians?
The question technically is:
Given a hyperelliptic curve $H/\overline{\mathbb{F}}_q$ and its jacobian $J_H$ of dimension $2$, if $...

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### Generalization of Area and Coarea formula for fractional Hausdorff measures

Let $X,Y$ be polish spaces, $s,t>0$ and $F:X\to Y$ locally Lipschitz continuous such that $X$ is $\sigma$-finite w.r.t. the $(s+t)$-dimensional Hausdorff measure $\mathcal{H}^{s+t}$.
The Eilenberg ...

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### Jacobians of genus 2 curves isogenous to a square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$

I would like to construct hyperelliptic curves whose Jacobians are isogenous to the square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$
My question is motivated by the following example.
...

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### Bounding the determinant of the Jacobian between a set and its polyhedral approximation

My question is, essentially, suppose I have two simply connected subset of $R^n$, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I ...

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### Resolution of the ideal of the Abel-Jacobi image of a curve?

Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I_{a(C)}$ whose terms are sums line bundles of the ...

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### Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix.
I've seen a similar ...

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### Are the Prym varieties geometrcally nondegenerate subvarieties of the Jacobians?

A subvariety $V$ of an abelian variety $X$ is geometrically nondegenerate if it meets any subvariety of $X$ of dimension bigger than or equal $codim(V)$.
My question is about the Prym varieties as ...

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### Polarization of the Prym variety

Let $X\rightarrow Y$ a ramifield double cover of curves, $J_X, J_Y$ their jacobians, $P\subset J_X$ the Prym variety, for any line bundle $L$ on $X$ of degree $g_X-1$, denote by $\Theta_L$ the ...

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### About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve

I was looking for some information related to the values of the characteristic polynomial $\chi(t)$ of the Frobenius of a Jacobian of a hyperelliptic curve $C$ of genus 2 over $\mathbb{F}_q$ and in ...

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### functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...

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270 views

### Is there a covering of Prym variety?

$\mathstrut$Hi, guys!
Let $C$, $C^\prime$ be projective smooth irreducible algebraic curves over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$), $\phi : C$ $\to$ $C^\prime$ a two-...

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### When stable curves can be embedded to their Jacobian?

Let $X$ be a stable curve over a complete DVR $R$ with smooth generic fiber. If $X$ has a $R$-rational point, by the universal properties of Neron models, we obtain a morphism from $f: X-X_{s}^{\rm ...

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### Do all simple factors of jacobians of curves come from correspondences?

For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over ...

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### Is there a concrete description of $\Theta_{sing}$ for a generic curve of genus 6?

If C is generic of genus 6, then $\Theta_{sing}$ is a smooth surface. Can anyone give me a reference or a hint as to what that surface might be. What are the numerical characteristics of this surface? ...

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### Distribution of Mordell–Weil ranks of higher genus curves

By "nice curve", I mean a smooth, projective, geometrically integral curve over $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Jac}{\operatorname{Jac}}\Q$ with at least one $\Q$-rational point. The Mordell–...