Jacobian of meromorphic curves

稳定尖点曲线上亚纯希钦纤维化的模空间

The meromorphic Hitchin fibration over stable pointed curves: moduli spaces

本文在稳定尖点曲线叠上构造了半稳定亚纯希格斯丛模空间的万有部分紧化，建立了容许节点谱覆盖下纤维与紧化雅可比簇的 BNR 对应类比，并研究了留数受幂零共轭类约束时的平坦固有希钦态射。

曲线上新型与一般型亚纯1-形式

New and General Type Meromorphic $1$-forms on Curves

本文通过显式构造在$\mathbb{P}^1$、椭圆曲线和超椭圆曲线上给出了一大族新型与一般型亚纯1-形式，并将其与Hurwitz实现问题联系，得到判定$\mathbb{P}^1$上1-形式新旧性的算法。

Deforming curves in jacobians to non-jacobians I: curves in $C^{(2)}$

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in the second symmetric power $C^{(2)}$ of $C$. More precisely, given a pencil $g^1_d$ of degree $d$ on $C$, let $X$ be the curve parametrizing pairs of points in divisors of $g^1_d$ (see the paper for the precise scheme-theoretical definition). We prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=4$ or $d=5$, dim$H^0 (g^1_5) = 3$ and $C$ has gen…

Generators of Jacobians of Hyperelliptic Curves

This paper provides a probabilistic algorithm to determine generators of the m-torsion subgroup of the Jacobian of a hyperelliptic curve of genus two.

On the locus of genus $3$ curves that admit meromorphic differentials with a zero of order $6$ and a pole of order $2$

The main goal of this article is to compute the class of the divisor of $\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of $Ω\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to $\mathcal{M}_g$ and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge-Chen-Gendron-Gr…

Deforming curves in jacobians to non-jacobians II: curves in $C^{(e)}$, $3\leq e\leq g-3$

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in symmetric powers $C^{(e)}$ of $C$ where $3\leq e\leq g-3$. More precisely, given a pencil $g^1_d$ of degree $d$ on $C$, let $X$ be the curve parametrizing divisors of degree $e$ in divisors of $g^1_d$ (see the paper for the precise scheme-theoretical definition). Under certain genericity assumptions on the pair $(C, g^1_d)$, we prove that if $X$ deforms infinitesimally out of the jacob…

Fine compactified Jacobians of reduced curves

To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compa…

Deformations of plane curves and Jacobian syzygies

We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.

关于节点曲线的精细紧致雅可比簇的分类

On the classification of fine compactified Jacobians of nodal curves

本文引入藤型精细紧致雅可比簇，刻画其作为光滑曲线雅可比簇极限及Néron模型紧致化的条件，并给出轨道偏序集的组合刻画。

On Jacobians of curves with superelliptic components

We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the Jacobian induced by the automorphisms of the curve. Moreover, we generalize a construction in \cite{Ya} of a family of non-hyperelliptic curves $\mathcal X_{r,s} $ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\mbox{Jac} (\mathcal X_{r, s})$ decomposes.

Singular curves and their compactified Jacobians

We survey the theory of the compactified Jacobian associated to a singular curve. We focus on describing low genus examples using the Abel map.

On the equivariant algebraic Jacobian for curves of genus two

We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.

A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree

Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree $p$, for $p$ a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain $p$-gonal curves.

广义费马曲线雅可比簇的分解

Decomposition of jacobians of generalized Fermat curves

本文研究了广义费马曲线雅可比簇的分解，扩展了Humbert-Edge曲线的相关研究成果，并增加了因子数量的计数。

Algebraic cycles on the Jacobian of a curve with a $g^r_d$

We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.

Computing branches and asymptotes of meromorphic functions

In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily compute the whole branch and all the generalized asymptotes of a special curve defined in n-dimensional space by a parametrization that is not necessarily rational. So, some new concepts and methods are established for this type of curves. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous paper…

阿贝尔曲面中的曲线对合及其雅可比簇

Involutions of curves in abelian surfaces and their Jacobians

该研究针对复阿贝尔曲面上型为$(1,d)$的极化线丛线性系统中的亏格二曲线平展覆盖，给出了曲线对合不动点计数及雅可比簇同源分解的相关结果。

On the Hilbert vector of the Jacobian module of a plane curve

We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.

Autoduality of compactified Jacobians for curves with plane singularities

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare line bundle. We prove that the Fourier-Mukai transform given by P is an auto-equivalence of the derived category of J'.

On the genus of curves in a Jacobian variety

We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For low values of g the second inequality can be further improved to p>2g-2.

On the approximate jacobian Newton diagrams of an irreducible plane curve

We introduce the notion of an approximate jacobian Newton diagram which is the jacobian Newton diagram of the morphism $(f^{(k)},f)$, where $f$ is a branch and $f^{(k)}$ is a characteristic approximate root of $f$. We prove that the set of all approximate jacobian Newton diagrams is a complete topological invariant. This generalizes theorems of Merle and Ephraim about the decomposition of the polar curve of a branch.

Restriction of stable bundles on a jacobian of genus 2 to an embedded curve

The aim of this note is to describe the restriction map from the moduli space of stable rank 2 bundle with small $c_2$ on a jacobian $X$ of dimension 2, to the moduli space of stable rank 2 bundles on the corresponding genus 2 curve $C$ embedded in $X$.

关于亏格二曲线雅可比簇的广义Kummer四重簇

On the generalised Kummer fourfold of the Jacobian of a genus two curve

本文构造了亏格二曲线雅可比簇的广义Kummer四重簇的双有理模型，利用三次曲线线性系统解释加法律，证明模型有轻微奇点并存在到四维射影空间的有限分歧覆盖。

Generators of Jacobians of Genus Two Curves

We prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve.

雅可比簇的商

Quotients of Jacobians

本文证明亏格≥5的曲线雅可比簇在有限群作用下的商不是单直纹的，并分类了亏格3和4中商为单直纹的所有曲线。

Computing isogenies between Jacobian of curves of genus 2 and 3

We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.

A correction to the paper "On Curves with Split Jacobian"

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the formula given in [5] works only under certain conditions.

Jacobian varieties with many elliptic curves

In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties. The first ones, provided by Ekedahl and Serre, are examples for which the isogenous decomposition has all factors being elliptic curves. It is well known that given two elliptic curves $E_{1}$ and $E_{2}$, there is a closed Riemann surface $X$ of genus two, with equations in terms of the elliptic curves, and whose jacobian variety $JX$ is isogenous to $E_{1} \times E_{2}$. In this paper, gi…

Simpson Jacobians of generalized tree-like curves

The compactified Jacobian of any projective curve $X$ is defined as the Simpson moduli space of torsion free rank one degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian in the case where $X$ is a generalized tree-like curve, i.e., a projective, reduced and connected curve such that the intersection points of its irreducible components are disconnecting ordinary double points. We prove that it is isomorphic to the product of the compactified Jacobians of a certain degree $d_…

Curves, Jacobians, and Cryptography

The main purpose of this paper is to give an overview over the theory of abelian varieties, with main focus on Jacobian varieties of curves reaching from well-known results till to latest developments and their usage in cryptography. In the first part we provide the necessary mathematical background on abelian varieties, their torsion points, Honda-Tate theory, Galois representations, with emphasis on Jacobian varieties and hyperelliptic Jacobians. In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and …

一类用于约化曲线族的新紧化 Jacobian

A new class of compactified Jacobians for families of reduced curves

本文为约化曲线族定义相对紧化 Jacobian，引入 V-稳定性条件进行构造，证明纤维为概形，并在平面奇点情形验证其保持经典对象的良好性质。

Stable maps to rational curves and the relative Jacobian

We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of Cavalieri-Marcus-Wise where the same identity was proved over on the locus of rational tails curves.

Locally symmetric families of curves and jacobians

This paper addresses the following question of Oort: "Are there any postive dimensional locally symmetric subvarieties of the moduli space of abelian varieties that are contained in the jacobian locus and contain the jacobian of at least one smooth curve? Some partial results are given.

Description of generalized jacobians of singular hyperelliptic curves through phase spaces of Mumford systems

Many finite dimensional integrable systems qre expressed with the help of the Lax equation which highlights a spectral parameter and therefore a spectral curve. These spectral curves are the starting point of an algebro-geometric investigation. The algebraic aspect of this investigation will be expressed using the Jacobians of smooth spectral curves and generalized jacobians of singular spectral curves and in the other hand the geometrical aspect will be attributed to the vector fields which are defined by the Lax equation. In this article will be dedicated only to singular hyperelliptic spect…

有理曲线邻域上亚纯函数域的稳定性

Stabilization of fields of meromorphic functions on neighborhoods of a rational curve

该研究针对含正自交光滑有理曲线的光滑连通复曲面，证明存在包含该曲线的邻域U，使得U内曲线连通邻域上的亚纯函数均可延拓至整个U，完善了复曲面亚纯函数延拓理论。

On total reality of meromorphic functions

We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image.

Decomposed Richelot isogenies of Jacobian varieties of curves of genus 3

For a nonsingular projective curve $C$ of genus 3 defined over an algebraically closed field of characteristic $p > 2$, we give a necessary and sufficient condition that the Jacobian variety $J(C)$ has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.

Integral points on plane curves and Plane Jacobian Conjecture over number fields

Let $K$ be a number field and $O_K$ the ring of integers of $K$. In the spirit of Siegel's theorem on integral points on affine algebraic curves, the plane Jacobian conjecture over $K$ is equivalent to the following statement: if $P,Q\in O_K[x,y]$ and $P_xQ_y-P_yQ_x\equiv 1$, then the curve $P=0$ has more than one integral point.

The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians

We give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\ZZ[\z_7]$.

Simpson Jacobians of reducible curves

For any projective curve $X$ let $\bar{M}^d(X)$ be the Simpson moduli space of pure dimension one rank 1 degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. When $X$ is a reduced curve the connected component of $\bar{M}^d(X)$ that contains semistable line bundles can be considered as the compactified Jacobian of $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian for the following projective curves: tree-like curves and all reduced and reducible curves that can appear as Kodaira singular fibers of an elliptic fibration,…

具有同构非极化雅可比簇的亏格2曲线对族

Families of Pairs of Genus 2 Curves with Isomorphic Unpolarized Jacobians

本研究基于Howe论文的注记，在域K上构造了三族由射影直线开子集参数化、具有同构非极化雅可比簇的亏格2曲线对，并指出该技术在Q上应用存在阻碍。

Constructing all Genus 2 Curves with Supersingular Jacobian

L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow \mathbb{P}^1$ of genus 2 curves with supersingular jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.

An explicit Construction of the Jacobian of a nonsingular curve and its group law

This paper is devoted to constructing an explicit efficient representation for the Jacobian variety of a nonsingular curve of genus greater than 1, and its group law. We describe an algorithm for executing the group law on the Jacobian elements, consisting of several steps. Finally we exhibit two examples. The first example exhibits an explicit formulation for the Jacobian of a well known non singular elliptic curve of genus 1, that is, the Weierstrass curve. The second example introduces an explicit computation of the group law for a non singular curve of genus 3, $C : ω_2^4 = ω_1 (ω_1-ω_0)(ω…

Decomposition of Jacobian varieties of curves with dihedral actions via equisymmetric stratification

Given a compact Riemann surface $X$ with an action of a finite group $G$, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety $JX$, known as the group algebra decomposition of $JX$. We consider the set of equisymmetric Riemann surfaces $\mathcal{M}(2n-1, D_{2n}, θ)$ for all $n\geq 2$. We study the group algebra decomposition of the Jacobian $JX$ of every curve $X\in \mathcal{M}(2n-1, D_{2n},θ)$ for all admissible actions, and we provide affine models for them. We use the topological equivalence of actions on the curves to obtain facts regarding its Jacobians. We …

Semistable modifications of families of curves and compactified Jacobians

Given a family of nodal curves, a semistable modification of it is another family made up of curves obtained by inserting chains of rational curves of any given length at certain nodes of certain curves of the original family. We give comparison theorems between torsion-free, rank-1 sheaves in the former family and invertible sheaves in the latter. We apply them to show that there are functorial isomorphisms between the compactifications of relative Jacobians of families of nodal curves constructed through Caporaso's approach and those constructed through Pandharipande's approach.

Cycles on curves and Jacobians: a tale of two tautological rings

We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber's Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question whether all tautological relations are motivic, giving a possible explanation why the Gorenstein properties…

Mult-Projective Models for Jacobian Varieties of Genus Two Curves

A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimension…

积分曲线与雅可比猜想

Integral curves and Jacobian Conjecture

本文针对雅可比猜想，基于L.M. Drużkowski的结果，结合半代数几何与向量场积分曲线的相关结论，推导出平展多项式自同态即为多项式自同构。

First steps towards total reality of meromorphic functions

It was earlier conjectured by the second and the third authors that any rational curve $g:{\mathbb C}P^1\to {\mathbb C}P^n$ such that the inverse images of all its flattening points lie on the real line ${\mathbb R}P^1\subset {\mathbb C}P^1$ is real algebraic up to a linear fractional transformation of the image ${\mathbb C}P^n$. (By a flattening point $p$ on $g$ we mean a point at which the Frenet $n$-frame $(g',g'',...,g^{(n)})$ is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic function…

Tautological rings on Jacobian varieties of curves with automorphisms

Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.

Non-Cyclic Subgroups of Jacobians of Genus Two Curves with Complex Multiplication

Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves with complex multiplication. In particular, we show that the Weil- and the Tate-pairing on such a Jacobian are non-degenerate over the same field extension of the ground field.

Geometry of the theta divisor of a compactified jacobian

We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.

Shimura curves within the locus of hyperelliptic Jacobians in genus three

We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…

关于平面曲线的Jacobian格式

On the Jacobian Scheme of a plane curve

本文研究平面曲线普通奇点的Jacobian格式，用初等Gröbner基方法计算Tjurina数至最小值，并给出基于Mather-Yau定理的二重点解析类型判定算法。