# Questions tagged [jacobians]

The jacobians tag has no usage guidance.

155
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### Why an elliptic curve can be defined as an abelian variety of dimension 1?

Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point&...

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### Does the Jacobian functor respect deformations?

I am trying to understand the relationship between deformations of curves and deformations of their Jacobians and would greatly appreciate a sanity check. Let $C_0$ be a smooth projective curve over a ...

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### What do we know about the $\mathbb{F}_{q^2}$-curve $X^{q-1} + Y^{q-1} + Z^{q-1}$?

People have thoroughly studied the Hermitian $\mathbb{F}_{q^2}$-curve $H: X^{q+1} + Y^{q+1} + Z^{q+1} = 0$. What do we know about the similar $\mathbb{F}_{q^2}$-curve $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ...

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### Intermediate Jacobian under group action

Let $X$ be a smooth Fano threefold with a finite group $G$ action. Assume that the orbit space $X/G$ is smooth. Is it true that $J(X/G)\cong J(X)^G$ As an abelian variety? Here, $J(X)^G$ is the $G$-...

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### How to prove a concentration isoperimetric inequality for a non-Lipschitz function

Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$,
\begin{align}
\...

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### Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection.
Can one understand how the compactified Jacobian of $C$ looks like?
For example, the ...

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### Under what conditions is the least-squares approximation bounded with the same Lipschitz gradient constants?

Let $f(x):\mathbb{R}^K\Longrightarrow \mathbb{R}^L$ denote a multivariate continuously differentiable function. All the partial derivatives of $f$ (all its Jacobian elements) are bounded from above ...

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### Properties of the quadratic form for the sum of the Jacobian and its transpose

Consider the Jacobian matrix of
$$
\begin{bmatrix}
f_{1}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))\\
\vdots \\
f_{N_2}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))
\end{bmatrix}
$$
Let the jacobian ...

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### Full-rank matrix

I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & ...

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### Elements of order two in a Prym variety

Let $Y_0$ be a genus two projective smooth complex curve,
let $Y_1$ be an étale cover of degree 2 of $Y_0$, and let
$\sigma$ be the involution of $Y_1$ over $Y_0$.
If $J_1$ is the jacobian of $Y_1$, ...

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### Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian

It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...

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### Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?

$\newcommand{\id}{\operatorname{Id}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TM}{\operatorname{TM}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Cof}{\operatorname{Cof}}
\newcommand{\Det}{\...

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### Derivative of dual isogeny is pullback on $H^1$

Apologies if this is not quite at the level of MathOverflow, but I have already asked at MSE and received no answer after two+ weeks and a bounty.
Let $X$ and $Y$ be elliptic curves (over an ...

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### Grassmannian of line subbundle of a stable rank 2 vector bundle on a smooth projective curve

Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $...

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### Why is the Jacobian of a curve "irreducible" as a principally polarized abelian variety?

In J.P. Murre's "Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of mumford", in the proof of Theorem 3.11 he remarks that "the Jacobian of a ...

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### Relation between the cohomology group of a curve and the cohomology group of its jacobian

Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that ...

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### How often is the rank of J_0(p)^- zero

As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...

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### Dual family of torsion-free rank-1 sheaves on Gorenstein curves

Let $X$ be a Gorenstein curve over a field an consider the compactified Jacobian parametrizing torsion-free, rank-1 sheaves on $X$.
Is there a chance that the dual functor $Hom(\_, \mathcal O_X)$ ...

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### Richelot isogenies in characteristic $2$

I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...

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### What is the involution on the moduli space of genus 3 curves induced by the Torelli map

Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli ...

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### Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)

Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...

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### An explicit equation for $X_1(13)$ and a computation using MAGMA

By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$.
And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(...

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### The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction

Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...

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### The intersection form on a Jacobian

$\DeclareMathOperator{\End}{End}$
Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the ...

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### Flat connection of a degree zero line bundle on curve

The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...

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### The image of a curve under the multiplication endomorphism of its Jacobian

Let $X$ be a complex smooth projective curve of genus $g\geq 2$. Embed $X$ in its Jacobian
${\rm{J}}(X)\cong{\rm{Div_0}}(X)/{\rm{Div_p}}(X)$ where ${\rm{Div_0}}(X)$ is the group of degree zero ...

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### The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization

Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...

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### Using principal polarisation to "cancel" Jacobian summands in isomorphism

I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces.
Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...

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### Embedding of a derived category into another derived category

I am considering the following two cases:
Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...

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### Continuations of derivations of Jacobian subring

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\...

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### Functions with a Jacobian whose columns are orthogonal

I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f_1(\mathbf{x}...

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### 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) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
(...

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### 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 ...

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### 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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### 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 ...

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### 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 \...

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### 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 (...

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### 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-...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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^{...

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### 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 ...

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### 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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### 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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### 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 = ...

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### 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 $(\mathbb{D}^n)^o$ ...

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### 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\...

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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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### 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 ...