Nguyen Van Chau
It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y]^2$ with $P_xQ_y-P_yQ_x \equiv 1$ has an inverse map in $\mathbb{Q}[x,y]^2$ if the fiber $P=0$ contains an infinite subset of $ d^{-1}\mathbb{Z}^2$ for an integer $d$.
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Using McMullen's rational surface automorphisms, we construct projective rational manifolds of higher dimension admitting automorphisms of positive entropy with arbitrarily high number of Siegel disks and those with exactly one Siegel disk.
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.
We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of dimension 2.
A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmueller modular forms. It turns out that for g>3 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply re…
Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counter examples to the SRJC a…
Jacobians with group actions and rational idempotents
本文的目的是证明有限群$G$的有理幂等元的一些一般性结果，并从这些结果中推导出具有$G-$作用的曲线的雅可比簇的分解中出现的分量的几何信息.我们给出了一个算法，以找到显式的原始有理幂等元的任何$G$，以及合理的投影仪在任何给定的子群不变。这些明确的建设允许几何描述的因素出现在分解的雅可比群作用：从他们我们推导出任何Prym或雅可比簇中间覆盖的分解，在雅可比的情况下，与$G-$行动。特别地，我们给出了中间覆盖的Prym簇是这样的因子的一个充要条件.
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.
On the tautological ring of a Jacobian modulo rational equivalence
本文研究了曲线雅可比行列式具有有理系数的Chow环。假设D是曲线的无基点g^r_d中的除数，使得典范除子K是除数D的倍数。我们发现重言式循环之间的关系。我们给出了具有p ^1的d次覆盖且其分歧点都是d阶的曲线的应用，然后给出了超椭圆曲线的应用。
Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-λ)\_λ$ is a rational polynomial, and if the Jacobian J(f,g) is a nonzero constant for some polynomial g in K[x,y], then K[f,g] =K[x,y].
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $g\geq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
Intermediate Jacobians and rationality over arbitrary fields
该研究建立任意域上几何有理三维簇的中间雅可比簇理论，证明域上两二次曲面的三维光滑完全交为k-有理等价于含k-定义直线，并得到首批在纯不可分扩张上有理但自身非k-有理的带k-点光滑射影簇例子。
We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.
A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.
Jacobian conjectures (that nonsingular implies invertible) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. The extensions for the Pinchuk counterexamples to the strong real Jacobian conjecture have no nontrivial automorphisms, but are of degree six. The birational case is proved, the Galois case is clarified but the general case of odd degree remains open. However, cert…
We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational point of order 48, parametrized by a rank-2 elliptic curve over Q, and we exhibit a single genus-2 curve over Q whose Jacobian has a rational point of order 70, the largest order known. We also give new examples of genus-2 Jacobians with rational points of order 27, 28, and 39. Most of our examples are produced by `gluing' two elliptic curves together along t…
Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then…
We show a direct calculation of Jacobian matrices in the old problems of rational curves on generic hypersurfaces.
Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Rieme…
Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is obtained by resolving this map. Several structural results are proven, revealing a rather orderly structure of the graph of the curves at infinity. We also exhibit and discuss a graph that may lead to a counterexample to the Jacobian Conjecture.
We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.
Plane Jacobian problem for rational polynomials
这篇论文已经撤回了作者由于一个关键的错误，在最后几行证明引理3.3。
We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.
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…
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regul…
The generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds are conjectured to satisfy a certain multiple cover formula. This conjecture is equivalent to Pandharipande-Thomas's strong rationality conjecture on the generating series of stable pair invariants, and its local version is enough to prove. In this paper, using Jacobian localizations and parabolic stable pair invariants introduced in the previous paper, we reduce the conjectural multiple cover formula for local curves with at worst nodal singularities to the case of local trees of smoo…
Euler number of the compactified Jacobian and multiplicity of rational curves
我们证明了具有局部平面奇点的有理曲线$C$的紧致化雅可比矩阵的Euler数等于在$C$的半泛变形的基础上$C$-常数层的重数.特别是，由丘，Zaslow和Beauville分配的K3曲面$S$上的有理曲线的多重性与$S$的稳定映射的模空间中的归一化映射的多重性一致。
A rational map $ϕ: \mathbb{P}_k^m \dashrightarrow \mathbb{P}_k^n$ is defined by homogeneous polynomials of a common degree $d$. We establish a linear bound in terms of $d$ for the number of $(m-1)$-dimensional fibers of $ϕ$, by using ideals of minors of the Jacobian matrix. In particular, we answer affirmatively Question~11 in arXiv:1511.02933v2.
We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.
A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g - 3$. Katz, Rabinoff and Zureick-Brown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, pr…
We prove by explicit example that the compactified jacobian can be nonreduced. The example is a rational space curve of arithmetic genus 4. This answers a question posed by Cyril D'Souza in 1979.
Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{α, β}$ defined over $K(α, β)$, corresponding to $\mathfrak p$, where $α$ and $β$ satisfy a quadratic $α^2+ b β^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{α, β}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(α, β)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{α, β}$. …
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptiona…
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.
Non-Cyclic Subgroups of Jacobians of Genus Two Curves
设E是定义在有限域上的椭圆曲线。Balasubramanian和Koblitz证明了：如果单位m_l的l次根不包含在基域中，则基域的一个域扩张包含m_l当且仅当E的l-扭点在同一个域扩张上有理。我们将这个结果推广到亏格为两条曲线的Jacobian。特别地，我们证明了在基域的相同域扩张上，Weil对和Tate对是非简并的.由此得到了超奇异亏格两条曲线的Jacobian的l-扭子群的完整刻画。特别地，我们证明了当l>3时，l-挠点在次数至多为24的域扩张上是有理的。
In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta-functions.
Intermediate Jacobians and linearizability
本文构建中间 Jacobian 挠子障碍的等变形式，并用于研究圆锥丛、二次曲面丛及 Fano 三维簇的可线性化性。
Algebraic cycles on the Jacobian of a curve with a $g^r_d$
我们提出了有理系数的循环之间的关系模代数等价的雅可比矩阵的曲线。这些关系取决于曲线所允许的线性系。它们是在重言式环中得到的，重言式环是包含曲线（的嵌入）的最小子空间，并且在交、庞特里亚金积和由同素性引起的拉回和下推的基本运算下是封闭的。
We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup Γ_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3\C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind.
It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford's classic "Lectures on Theta", where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, …
Existence of arithmetic degrees for generic orbits and dynamical Lang-Siegel problem
本文证明支配有理自映射在一般轨道点处算术次数存在，并推广至拟射影簇上的 étale 态射，同时将结果用于动力 Lang–Siegel 问题，刻画局部高度增长与轨道子集的 Banach 密度。
Rational curves and the Hilbert Property on Jacobian Kummer varieties
本文在特征零的有限生成域上，证明了与亏格2曲线的Jacobian相关联的所有Kummer曲面具有Hilbert性质，并推广证明与奇数次亏格≥2超椭圆曲线相关联的所有Jacobian Kummer簇也具有该性质。
We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…
Jacobians among Abelian threefolds: a formula of Klein and a question of Serre
设k是一个域，f是加权h \geq 0且亏格g>1的Siegel模形式。利用f，我们定义了g维主极化阿贝尔簇（A，a）/k的k-同构类的一个不变量.此外，当（A，a）是一个光滑的平面曲线的雅可比矩阵，我们展示了如何关联到f一个经典的平面不变量。作为这些构造的直接结果，当g=3且k是复域的一个子域时，我们得到（i）Klein将模形式χ 18}与平面四次数判别式的平方联系起来的公式的一个新证明;（ii）通过观察（A，a）处的χ 18}的值是否是k中的平方，可以判定（A，a）何时是k上的Jacobian的一个证明。这回答了J的一个问题。P. Serre。最后，我们研究了这种方法的可能推广为g>3。
Let V be a rational quadratic space of signature (m,2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m+1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimed…
Holomorphic Eisenstein series with Jacobian twists
对模曲线$X_0(l)$的雅可比矩阵上的每一点，定义并研究了一类扭全纯Eisenstein级数.这些是扭曲模形式的更一般概念的特殊情况，它们对应于通常模形式的线丛的零度扭曲的模曲线$X_1(l)$上的截面。我们猜想雅可比矩阵上的一个点是有理的当且仅当这些相同权的扭曲爱森斯坦级数的比具有有理系数。
We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one member has all $δ$-invariants (resp. Jacobian numbers) strictly less than $(p-1)/2$ (resp. $p$), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobi…
The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$
研究 $\mathcal A_5$ 中 Siegel 度量限制在三次三维流形的中间雅可比轨迹上的第二基本形式，证明其像包含于某个乘法映射的核中。
Genus-One Fibrations and the Jacobian of Linear Slices in the Quintic Equal-Sum Problem
本文研究五次等和方程在特定约束下的解结构，证明了雅可比纤维的秩界及非奇异特性的条件。
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let Λbe its value group. Given a smooth, proper, connected K-curve X and a skeleton Γof the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(Γ) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classic…
Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves
本文确定亏格 2 曲线上秩 2 半稳定丛泛模空间的有理 Chow 环，证明其由重言类生成，并顺带得到相关泛 Jacobian 的 Chow 环。
On the local geometry of the moduli space of $(2,2)$-threefolds in ${\mathcal A}_9$
研究 ${\mathbb P}^2 \times {\mathbb P}^2$ 中 $(2,2)$-三维簇中间雅可比模空间的局部几何，证明 Siegel 度量第二基本形式与乘法映射的复合为非零全纯截面，并描述其核。
Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian
我们展示了如何为每一个整数n一个可以明确地构建n个不同的平面四次线和一个超椭圆曲线的复数所有的雅可比矩阵是同构的另一个阿贝尔簇没有极化。当我们说曲线可以“显式地”构造时，我们的意思是曲线的定义方程的系数是R中代数数的简单有理表达式，其在Q上的极小多项式可以精确地给出，并且其十进制近似值可以给出到尽可能多的位置，以将它们与它们的共轭区分开来。我们还证明了一个简单的陈述定理，允许一个决定是否两个平面四次线的复数，每对交换对合，是同构的。
Non-Cyclic Subgroups of Jacobians of Genus Two Curves with Complex Multiplication
设E是定义在有限域上的椭圆曲线。Balasubramanian和Koblitz证明了：如果单位m_l的l次根不包含在基域中，则基域的一个域扩张包含m_l当且仅当E的l-扭点在同一个域扩张上有理。本文将这一结果推广到亏格为两条曲线的具有复乘的Jacobian。特别地，我们证明了在相同的基域的域扩张上，这样的雅可比矩阵上的Weil配对和Tate配对是非退化的.
Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of discriminant $D$ over $\mathbb Q$ such that the Rosati involution coincides with a positive involution of the form $α\mapstoμ^{-1}\overlineαμ$ on $\mathcal O$ for some $μ\in\mathcal O$ with $μ^2+D=0$. In this paper, we first give a formula for the number of irreducible components in $\mathcal Q_D$, strengthening an earlier result of Rotger. Then for each irr…
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points ov…
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。
Quadratic torsion orders on Jacobian varieties
研究有理数域上亏格\(g\geq2\)超椭圆曲线，通过引入特定多项式族方法，证明其雅可比簇存在特定阶有理挠点，确定了不同\(N\)值下超椭圆曲线与有理挠点的关系。
Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC, for instance those that induce rational function field extensions of a given fixed odd degree.
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