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Questions tagged [jacobians]

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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&...
HaomengXu's user avatar
5 votes
1 answer
347 views

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 ...
Catherine Ray's user avatar
3 votes
1 answer
257 views

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} = ...
Dimitri Koshelev's user avatar
1 vote
1 answer
77 views

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 $G$-...
user41650's user avatar
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1 vote
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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} \...
XYZ's user avatar
  • 79
4 votes
1 answer
213 views

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 ...
IntegrableSystemsEnthusiast's user avatar
2 votes
0 answers
39 views

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 ...
Yarden Levy's user avatar
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0 answers
40 views

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 ...
Atharva's user avatar
2 votes
2 answers
212 views

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 & ...
MDR's user avatar
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1 answer
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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 $Y_1$, ...
user95246's user avatar
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6 votes
1 answer
353 views

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-...
TCiur's user avatar
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9 votes
0 answers
283 views

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}{\...
Asaf Shachar's user avatar
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4 votes
0 answers
151 views

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 ...
Hank Scorpio's user avatar
2 votes
1 answer
92 views

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 $...
hennlu's user avatar
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179 views

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 ...
TCiur's user avatar
  • 381
7 votes
1 answer
577 views

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 ...
Roxana's user avatar
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4 votes
0 answers
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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 ...
Maarten Derickx's user avatar
3 votes
1 answer
143 views

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 $Hom(\_, \mathcal O_X)$ ...
Raffaele C's user avatar
3 votes
0 answers
124 views

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, ...
Dimitri Koshelev's user avatar
3 votes
1 answer
257 views

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 ...
Asvin's user avatar
  • 7,302
3 votes
1 answer
262 views

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 - \...
AVP82000's user avatar
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4 votes
1 answer
287 views

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(...
k.j.'s user avatar
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4 votes
0 answers
77 views

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 ...
E. Kaya's user avatar
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2 votes
0 answers
53 views

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 ...
RandomMathUser's user avatar
1 vote
1 answer
219 views

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 ...
MKR's user avatar
  • 93
5 votes
0 answers
142 views

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 ...
KhashF's user avatar
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1 vote
0 answers
122 views

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 $\...
Aoki's user avatar
  • 297
3 votes
0 answers
107 views

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 \...
mathphys's user avatar
  • 285
12 votes
1 answer
491 views

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 $1$($\...
user41650's user avatar
  • 1,696
1 vote
1 answer
50 views

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}\...
A.Skutin's user avatar
  • 309
3 votes
1 answer
318 views

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}...
oasis93's user avatar
  • 33
4 votes
1 answer
167 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) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
user237522's user avatar
  • 2,613
10 votes
1 answer
334 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 ...
Yoav Len's user avatar
  • 137
2 votes
0 answers
60 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 ...
Fernanda's user avatar
1 vote
0 answers
106 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 ...
Asvin's user avatar
  • 7,302
2 votes
1 answer
355 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 \...
Watson's user avatar
  • 1,682
4 votes
3 answers
614 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 (...
Asvin's user avatar
  • 7,302
0 votes
0 answers
251 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-...
goh's user avatar
  • 101
3 votes
0 answers
147 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 ...
Chen's user avatar
  • 1,583
1 vote
1 answer
557 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 ...
a06e's user avatar
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5 votes
0 answers
146 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 ...
Jef's user avatar
  • 1,319
0 votes
1 answer
42 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^{...
Craig's user avatar
  • 105
2 votes
0 answers
118 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 ...
Libli's user avatar
  • 6,756
2 votes
1 answer
161 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 ...
Dimitri Koshelev's user avatar
0 votes
0 answers
79 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 -\...
User_8088's user avatar
1 vote
0 answers
61 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 = ...
Dimitri Koshelev's user avatar
4 votes
0 answers
79 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 $(\mathbb{D}^n)^o$ ...
Asaf Shachar's user avatar
  • 6,499
5 votes
1 answer
160 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\...
Eduardo R. Duarte's user avatar
5 votes
1 answer
179 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 $J$...
Ceka's user avatar
  • 501
3 votes
0 answers
137 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 ...
user237522's user avatar
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