C. -L. Chai, F. Oort
We show that for any family of curves over a base scheme of finite type over the prime field $\mathbb F_p$ such that the monodromy is ``maximal'', there exist infinitely many closed points of the base scheme such that the Jacobian of fibre is absolutely simple.
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We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t \in K$ and the Galois group of the polynomial $h(x)$ over $K$ is "very big" and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)
Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers math/0610138 [math.AG] and math/0605028 [math.AG] .
A Note on Jacobians with Prescribed Factors
本文在无限域上改进了Matsusaka定理，使得补同源因子可以是任意大维数的绝对单因子，并讨论了相对单性与同源Schottky问题的相关问题。
We prove that the jacobian of a hyperelliptic curve y^2=f(x) is absolutely simple if deg(f)=q+1 where q is a power prime congruent to 5 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois group is L_2(q).
Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians due to Zarhin to give many examples of higher dimensional, absolutely simple Jacobians over $k(t)$ with bounded rank in towers $k(t^{1/p^r})$. In many cases we are able to compute the rank at every layer of the tower.
Del Pezzo surfaces of degree 1 and jacobians
本文借助次数为1的Del Pezzo曲面构造非超椭圆4亏格曲线的绝对单Jacobian簇，是作者前期工作math.AG/0405156的延续。
Let $m \geqslant 6$ be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to $C_2^2 \rtimes_2 C_m$ admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
本文验证了某些亏格为2且雅可比簇绝对简单的Atkin-Lehner商曲线的强BSD猜想，通过证明其Shafarevich-Tate群平凡。
This paper is concerned with some Algebraic Geometry codes on Jacobians of genus 2 curves. We derive a lower bound for the minimum distance of these codes from an upper "Weil type" bound for the number of rational points on irreducible (possibly singular or non-absolutely irreducible) curves lying on an abelian surface over a finite field.
Explicit construction of decomposable Jacobians
本文给出了特征0域上可分解超椭圆 Jacobian 簇的显式构造，并应用于构造具有无穷多二次扭转且含有有理非 Weierstrass 点的超椭圆曲线族。
These are notes related to a 12-hour course of lectures given at the Centre de Recerca Mathemàtica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank. They may be viewed as a continuation of my Park City notes (arXiv:1101.1939). In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of higher genera and results over more ge…
We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple Jacobians. The third family consists of pairs of genus-3 curves, one member of each pair being a…
We give a shorter proof to a recent result by Neuberger, in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results presented in Neuberger.
In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+δ)$ where $δ$ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension $\leq 9$ in the case $\prod_{i=1}^{n}a_{ii}\neq0$.
By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study Mather-Jacobian multiplier ideals on algebraic curve, in which case the definition of Mather-Jacobian multiplier ideal can be extended to a ground field of any characteristic. We show that Mather-Jacobian multiplier ideal on curves is essentially the same as an integrally closed ideal. Finally by comparing conductor ideal with Mather-Jacobian multiplier ideal, …
A note on the real Jacobian conjecture in degree 7
该研究针对7次实雅可比猜想展开，证明了最高次齐次部分为特定形式的7次多项式对应的实雅可比猜想成立，同时排除了一类非典型雅可比对的存在性。
It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C under multiplication by n on JC. We prove that the subgroup generated by these cycles, in the Chow group modulo algebraic equivalence, has rank at most d-1, where d is the gonality of C. We also discuss some general facts on the action of n_* on the Chow groups.
We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill-Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of these invariants is determined by the other two.
This is a short addendum to a note of Beauville on the subject of the title. We prove an inequality that takes into account the constant part of the Jacobian.
Products of hyperelliptic Jacobians with maximal Galois image
本文研究了超椭圆 Jacobian 乘积的相伴 adelic 表示，给出了确保其具有极大 Galois 像的简单准则，并提出了一种构造 Jacobian 乘积使其除法域尽可能互不相交的方法。
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.
Let $F:\Bbb C^n\to\Bbb C^n$ be a polynomial mapping with a non vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, the set $S_F$ can not be connected (this is the Nollet-Xavier Conjecture). Additionally, if $n=2$, then the set $S_F$ of non-properness of $F$ cannot be a curve without self-intersections.
The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.
Rational torsion on hyperelliptic jacobian varieties
本文证明了对于特定范围内的 N，存在超椭圆雅可比变元家族具有有理 torsion 点，并给出了具体应用实例。
A note on multisecants of the Kummer variety of a Jacobian
本文研究光滑射影曲线上 $\mathfrak{g}^{n}_{2n}$ 诱导的有理映射，将其纤维与雅可比簇的库默尔簇的 Gunning 多割线建立联系，从而推广了第一作者与 Codogni、Salvati Manni 的已有结果。
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.
Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians
该研究比较二维曲线的三种高度，定义加权内隆-塔特高度并推导相关不等式，构造带无限阶截面的半稳定二元六次式族，应用得到Q上正秩二维雅可比簇等结果。
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 $X$.
Let A be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over the algebraic closure of k. There exists a canonical extension k' of k, of degree 1 or 2, such that A becomes isomorphic to a Jacobian over k'. The aim of this note is to give a geometric construction of this extension.
In this note we investigate the connection between polylogarithms on curves and abelian schemes. The main result shows that the polylogarithm on the abelian scheme can be obtained as the push-forward of the polylogarithm on a suitable sub-curve. If the abelian scheme is the Jacobian of a smooth projective curve, this push-forward can also be written as a cup-product with the fundamental class of the curve.
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.
We construct some natural cycles with trivial regulator in the higher Chow groups of Jacobians. For hyperelliptic curves we use a criterion due to J. Lewis to prove that the cycles we construct are indecomposable, and then use a specialization argument to prove indecomposability for more general curves. Please note that math.AG/9909062 has become an obsolete version of some of the results in the present paper.
Finding the complement of an elliptic curve inside a Jacobian
本文针对亏格2曲线X到椭圆曲线E的非平凡映射，给出一个简单算法，用于计算X的雅可比簇分解中互补的椭圆曲线因子。
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.
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。
Stable maps, multiplicities, and compactified Jacobians
研究复杂积分曲线的稳定映射、变形空间和紧化雅可比簇之间的数值关系，修正Fantechi-Göttsche-van Straten关于变形空间δ-常数层在[C]处乘积性的陈述，并给出原主张成立的充要条件。
On the Jacobian variety of the Accola-Maclachlan curve of genus four
本文研究四亏格 Accola-Maclachlan 曲线的 Jacobian 簇，显式给出其 Poincaré 同源分解，证明其同构于两个单 Abel 曲面的乘积并分别提供 Riemann 矩阵。
In this note, we examine the Jacobian ring description of the Hodge structure of zero loci of vector bundle sections on a class of ambient varieties. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [9] for projective spaces. As an additional application, we prove the Hodge conjecture for very generi…
Sections of Jacobian fibrations over lines
本文研究光滑曲面线性系中直线对应的通用雅可比截面的上同调类，针对K3曲面的通用紧化雅可比，修正了排印错误与猜想7，探讨结果与文献步骤的协调问题。
We calculate a Griffiths-type ring for smooth complete intersection in Grassmannians. This is the analogue of the classical Jacobian ring for complete intersections in projective space, and allows us to explicitly compute their Hodge groups.
Brill-Noether loci on an Enriques surface covered by a Jacobian Kummer surface
针对被 Jacobian Kummer 曲面覆盖的 Enriques 曲面，本文证明其特定秩 2 稳定向量丛模空间中存在真第一 Brill-Noether 轨迹。
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao's conjecture on the freeness of a line arrangement being determined by its combinatorics.
In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.
Complete verification of strong BSD for many modular abelian surfaces over $\mathbf{Q}$
本文发展理论与算法，无条件精确验证了Q上绝对简单模阿贝尔簇的强BSD猜想，涵盖28个亏格2的Atkin-Lehner商、97条LMFDB曲线及6条Wang曲线，首次确认维数≥2的绝对简单阿贝尔簇的强BSD，并验证Tate-Shafarevich群阶为7²。
Curves with Jacobian syzygies of the same degree
本文研究雅可比合系分次模由极小次数分量生成的复射影平面曲线，发现该类曲线规模庞大，包含光滑曲线、极大Tjurina曲线及满足特定三重点条件的直线构形。
Inverse limits of automorphisms of truncated polynomials and applications related to Jacobian conjecture
本文利用特征 $p$ 多项式环与逆极限技术，证明满足 Jacobian 条件且保持极大理想的多项式环同态必为自同构，为 Jacobian 猜想提供新结论。
Let $f_1,...,f_n$ be $n$ germs of holomorphic functions at the origin of $\mathbb{C}^n$ such that $f_i(0)=0$, $1\leq i\leq n$. We give a proof based on the J. Lipman's theory of residues via Hochschild Homology that the Jacobian of $f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ if and only if the dimension ot the germ of common zeos of $f_1,...,f_n$ is sttrictly positive. In fact we prove much more general results which are relatives versions of this result replacing the field $\mathbb{C}$ by convenient noetherian rings $\mathbf{A}…
We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are especially useful for estimating the class number of function fields in towers of function fields over finite fields. We give examples in the case of asymptotically good towers. In particular we estimate the class number of function fields which are steps of towers having one or several positive Tsfasman-Vladut invariants. Note that the study is not done as…
Arbitrarily large $p$-torsion in Tate-Shafarevich groups
本文通过构造违反 Hasse 原理的显式 $\mu_p$-覆盖，证明任意素数 $p$ 下，$\mathbb{Q}$ 上绝对单纯 Abel 簇的 Tate-Shafarevich 群可具有任意大的 $p$-挠群。
The purpose of these notes is to provide the details of the Jacobian ring computations carried out in [1], based on the computer algebra system Magma [2].
In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.
Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\char(K)=0$ and that the characteristic of the residue field is not 2. Suppose that the Jacobian $\Jac(C)$ has semi-stable reduction over $R$. Embed $C$ in $\Jac(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique moderately ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\Jac(C)$.
Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for instance it can be used to determined whether $C$ is free or nearly free. In this note we study the relations of this invariant $mdr(f)$ with a decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer i…
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