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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&...
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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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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 $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ... 1 vote 1 answer 77 views ### Intermediate Jacobian under group action Let$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$-... 1 vote 0 answers 123 views ### 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$t \geq 0, \begin{align} \... 4 votes 1 answer 213 views ### Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection SupposeC$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 ... 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 ... 0 votes 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 ... 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 & ... 2 votes 1 answer 132 views ### 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, ... 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-... 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}{\... 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 ... 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 ... 3 votes 0 answers 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 ... 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 ... 4 votes 0 answers 132 views ### 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 ... 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) ... 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, ... 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 ... 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 - \... 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(... 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 ... 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 ... 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 ... 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 ... 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 \... 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 \... 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(\... 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}\... 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}... 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\}. (... 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 ... 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 ... 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 ... 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 \... 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 (... 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-... 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 ... 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 ... 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 ... 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^{... 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}$has ... 2 votes 1 answer 161 views ### What is the quotient$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$0$) $$E\!:y_1^2 ... 0 votes 0 answers 79 views ### Isomorphism between 2nd symmetric product and Jacobian Let X=X_0(N) be hyperelliptic with g(X)\geq 2 with \infty as a cusp and \iota as the hyperelliptic involution. Then the map$$X^{(2)} \longrightarrow Jac(X)D \longrightarrow [D-\infty -\... 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 = ... 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$(\mathbb{D}^n)^o$... 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\...
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 $J$...
### 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 ...