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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0answers
55 views
$A,B,w \in \frac{\mathbb{Q}[t]}{(t^2-1)}[x,y]$: $\operatorname{Jac}(A,B)=1$, $\operatorname{Jac}(A,w)=0$, $w \notin \frac{\mathbb{Q}[t]}{(t^2-1)}[A]$
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 $A,B,w \in R[x,y]$ ...
3
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0answers
120 views
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 ...
5
votes
1answer
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) ...
6
votes
1answer
236 views
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 ...
4
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1answer
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 ...
1
vote
0answers
94 views
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 ...
2
votes
1answer
181 views
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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votes
2answers
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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votes
4answers
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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....
3
votes
1answer
216 views
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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0answers
251 views
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 $Aut(Jac(X', a'))$ ...
5
votes
1answer
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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0answers
102 views
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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0answers
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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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0answers
125 views
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)...
4
votes
1answer
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 $[...
3
votes
2answers
320 views
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 $...
3
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0answers
136 views
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 $\mathrm{NS}(J)\cong\mathbb{Z}\cdot\theta$? ...
3
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1answer
192 views
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,...
6
votes
1answer
208 views
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 $f,g \in k[h]$,
...
5
votes
2answers
178 views
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 ...
3
votes
0answers
121 views
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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0answers
126 views
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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1answer
164 views
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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0answers
207 views
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 ...
3
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1answer
173 views
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 $[...
3
votes
1answer
143 views
5-Descent or ($\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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0answers
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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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0answers
131 views
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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2answers
218 views
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 ...
5
votes
1answer
170 views
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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0answers
125 views
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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0answers
191 views
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/...
3
votes
1answer
240 views
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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1answer
134 views
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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0answers
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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 ...
5
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1answer
261 views
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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0answers
61 views
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 ...
3
votes
1answer
198 views
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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2answers
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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 ...
3
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1answer
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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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0answers
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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 ...
3
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1answer
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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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2answers
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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 = \...
4
votes
1answer
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-...
0
votes
1answer
182 views
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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1answer
422 views
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 ...
4
votes
0answers
189 views
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? ...
8
votes
1answer
368 views
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–...
