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2
votes
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98 views

Certain endomorphisms of $\mathbb{C}(x,y)$

Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$ satisfying the following two conditions: (i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
7
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1answer
192 views

Surjectivity of the Abel-Prym map

It is well known that the Abel-Jacobi map restricted to $\text{Eff}_g(C)$ surjects onto the Jacobian $\text{Jac}(C)$, since every divisor of degree $g$ is effective. Is there an analogous statement ...
2
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0answers
28 views

Calculate the Jacobian of a particular diffeomorphism of parallelizable manifold onto itself

Let $M$ be $d$-dimensional parallelizable manifold. Let $e_k(x)$ for $k=1, \dots , d$ be the smooth vector fields forming orthonormal basis in tangent bundle of 𝑀, these vector fields exist because ...
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0answers
52 views

A map on Jacobians coming from a correspondence explicitly

From this question, we know that every map of the form $J(C) \to J(C)$ for a curve $C$ and it's jacobian $J(C)$ comes from a correspondence between $C\times C$ and in fact we can take this ...
2
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1answer
182 views

Degree of morphisms and isogenies

$\renewcommand{\J}{\mathrm{Jac}} \renewcommand{\F}{\mathbb{F}}$ I am reading this paper by B. Gross, and there is something I don't understand on p. 945. Here is the context: fix a prime $p \equiv 3 \...
3
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3answers
462 views

Torsion in the jacobian of a super elliptic curve

Let $y^n = f(x)$ define a smooth projective curve $C$ over some field $k$ with $\deg f \geq n$ and odd and with $f(x)$ having no repeated roots. Let $J$ be the Jacobian of $C$ and $J[n]$ it's (...
0
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0answers
92 views

What is the jacobian of an image lookup function?

I posted this question on Robotics Stack Exchange (link) but thought it could be relevant here as well. I'm trying to solve a computer vision problem whereby I wish to use Levenberg–Marquardt non-...
3
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0answers
117 views

non-singular divisors of the jacobian variety

Let $X$ be a smooth, projective curve of genus at least $4$. The well-known divisor $\theta$ of the associated Jacobian variety is $\mathrm{Jac}(X)$ is singular and also ample. The $\theta$ divisor ...
1
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1answer
76 views

Jacobian of changing of variables to singular value decomposition

It is well known that changing variables from a symmetric matrix to its eigenvalue decomposition involves a Jacobian which is just the Vandermonde determinant of the eigenvalues. Now suppose I have a ...
5
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0answers
105 views

Unirationality of universal Jacobian over special strata of moduli space of pointed genus 3 curves

Let $M_{3,1}$ be the (coarse, non-compactified) moduli space of genus $3$ curves with a marked point over a field $k$ of characteristic zero. Throwing away the hyperelliptic curves, take the open ...
0
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1answer
32 views

Keeping the covariant divergence intact under changes of frame

In Eulcidean 3-space with coordinates $(r, \theta, \phi)$ where $\theta$ is the polar angle and $\phi$ the azimuthal angle, we may write the covariant divergence of a vector $E = E^\mu e_\mu$ as $$E^{...
2
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0answers
87 views

Weak Lefschetz property Jacobian ring smooth hypersurface

Let $A_{.}$ be a graded commutative ring. We say that $A_{.}$ satisfies the weak Lefschetz property if for generic $L \in A_1$ the multiplication maps $ \times L : A_i \longrightarrow A_{i+1}$ has ...
2
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1answer
135 views

What is the quotient $E \!\times\! E^\prime / G$?

Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 ...
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0answers
64 views

Isomorphism between 2nd symmetric product and Jacobian

Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map $$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...
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0answers
47 views

Is there a non-singular irreducible genus $2$ curve $C/\mathbb{F}_p$ and two $\mathbb{F}_p$-coverings $C \to E$, $C \to E^{(1)}$ of some degree $n$?

Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 = ...
4
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0answers
65 views

Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

Let $\mathbb{D}^n$ be the closed unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; More precisely, I assume that $f$ is real-analytic and harmonic on the interior $(\mathbb{D}^n)^o$ ...
5
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1answer
142 views

Identifying elements in the kernel of an explicit endomorphism of a Jacobian variety

I hope this question fits here. Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x_1,y_1)+(x_2,y_2)-2\...
4
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0answers
101 views

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 $J$...
3
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0answers
129 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
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1answer
222 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
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1answer
259 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 ...
5
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1answer
470 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
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0answers
115 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
256 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 ...
7
votes
2answers
455 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 ...
23
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4answers
1k views

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....
2
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1answer
244 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 ...
12
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0answers
276 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
193 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 ...
7
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0answers
169 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/...
2
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0answers
103 views

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 ...
8
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0answers
141 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)...
5
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1answer
365 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
444 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
161 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 $\mathrm{NS}(J)\cong\mathbb{Z}\cdot\theta$? ...
4
votes
1answer
285 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
216 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
193 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
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0answers
138 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 ...
1
vote
0answers
174 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 $...
4
votes
1answer
174 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 ...
6
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0answers
233 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
votes
1answer
202 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
163 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 ...
4
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0answers
83 views

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 \...
5
votes
0answers
158 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 ...
6
votes
2answers
246 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
226 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 + ...
2
votes
0answers
274 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 \...
2
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0answers
376 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/...

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