Questions tagged [jacobians]
The jacobians tag has no usage guidance.
53
questions with no upvoted or accepted answers
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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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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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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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Complete curves in $\mathcal{M}_g$ all of whose Jacobians have trivial endomorpism ring
I'm trying to construct complete smooth curves $C$ in $\overline{M}_g$ such that for all points $S \in C \cap \mathcal{M}_g$, its Jacobian $\text{Jac}(S)$ satisfies $\text{End}(\text{Jac}(S)) = \...
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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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Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity
Let $X$ be a hyperelliptic curve of genus $g \ge 2$ over a field $k$ (of characteristic not 2, 3 or 5, if you like, but could be positive in general). Let $J$ be the Jacobian of $X$, thought of as $\...
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An example computation of etale cohomology
(edited for clarity)
In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
- 13k
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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 ...
- 1,801
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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 ...
- 2,817
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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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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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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:
$$\delta f (x) = \text{convex hull} \left \{\lim_{x_i \...
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Derivative of dual isogeny is pullback on $H^1$
Apologies if this is not quite at the level of MathOverflow, but I have already asked at MSE and received no answer after two+ weeks and a bounty.
Let $X$ and $Y$ be elliptic curves (over an ...
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How often is the rank of J_0(p)^- zero
As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...
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The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction
Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...
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Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?
Let $\mathbb{D}^n$ be the closed unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; More precisely, I assume that $f$ is real-analytic and harmonic on the interior $(\mathbb{D}^n)^o$ ...
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How to describe the subspace of invariants under the Rosati involution?
Consider the Jacobian $J_C$ of hyperelliptic curve
$$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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Is there a concrete description of $\Theta_{\mathrm{sing}}$ for a generic curve of genus $6$?
If $C$ is a generic curve of genus $6$, then $\Theta_{\mathrm{sing}}$ is a smooth surface. Can anyone give me a reference or a hint as to what that surface might be. What are the numerical ...
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Computing Tamagawa numbers for jacobians of hyperelliptic curves
Do exist some computational approach to calculation of Tamagawa number for the jacobian of hyperelliptic curve at prime $p$?
As followed from this question one can compute $\Phi(\overline{\mathbb F}...
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Jacobians - intuitive proof required for change of variable in integration
I understand the need for a factor to account for change of units of length/area/volume in multiple integration up to triple integration - & understand why the Jacobian is the appropriate factor, ...
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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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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, ...
- 1,801
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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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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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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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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 $\mathrm{NS}(J)\cong\mathbb{Z}\cdot\theta$? ...
user39380
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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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Endomorphisms of Jacobians of Hyperelliptic Curves taking Exceptional Divisors to Exceptional Divisors
I am trying to find an hyperelliptic curve (say of genus 2) together with an endomorphism $\phi$ of its Jacobian with the property that it sends Mumford reduced divisors of the exceptional form $P-\...
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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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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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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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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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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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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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Which curves cut the Hyperelliptic locus?
Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of ...
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Curve C of genus 2 whose equation satisfies equation in Igusa invariants, but where Jac(C) does not split
The background question is: Let $C$ be a curve of genus $2$ over a field $k$. When is there a degree $2$ morphism from $C$ to an elliptic curve (and therefore an isogeny from the Jacobian $\mbox{Jac}(...
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Reference for Jacobians in characteristic $p$
I am looking for a basic reference for Jacobians of algebraic curves in characteristic $p>0$. I just want basic facts about the relation between the curve and its Jacobian.
I dont want to assume ...
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deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
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Why is the Eisenstein quotient a quotient of the new part of the Jacobian?
Dear MO Community,
Let $X = X_0(N)_{/\mathbb{Q}}$, and $J$ its jacobian. Mazur defines the Eisenstein quotient of $J$, denoted $\widetilde{J}$, as
\[ 0 \rightarrow \gamma_IJ \rightarrow J \...
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Numerical partial differentiation of a convolution product with FFT
How can one numerically calculate the partial derivatives of a convolution function, particularly when the closed-form or analytical expressions of the derivatives are not readily available? I am ...
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Numerical estimation of partial derivatives of convolved functions when closed forms do not exist
Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
- 803
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Cokernel of the Jacobian Matrix
For an algebraic variety $V = \mathcal{V}(f_1,\dots,f_m)\subset \mathbb{C}^n$, in smooth points $p$ there is a nice geometric interpretation of the Jacobian $(\partial f_i/\partial x_j)_{ij}\lvert_p$'...
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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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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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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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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 = ...
- 1,801
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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 ...
- 2,022
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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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Inverse Transpose of Jacobian Matrix
Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by
\begin{equation}
f(x)\approx f(...
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Abel-Jacobi map for regular fibered surfaces.
Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let $...
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