Jorge Alberto Guccione, Juan José Guccione, Rodrigo Horruitiner, Christian Valqui
We translate the results of Yansong Xu into the language of~\cite{GGV1}, obtaining nearly the same formulas for the intersection number of Jacobian pairs, but with an inequality instead of an equality.
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In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair $(X,Z)$ where $X$ is a smooth complete intersection in the projective space and and $Z$ is a simple normal crossing divisor in $X$ whose irreducible components are smooth hypersurface sections on $X$. Our Jacobian rings give an algebraic description of the cohomology of the open complement $X-Z$ and it is a natural generalization of the Poincaré residue representation of the cohomology of a hypersurface originally invented by Griffiths. The main results generalize the Macaulay's duality the…
Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A relative polar arc is a Newton-Puiseux root, $x=γ(y)$, of the Jacobian $J=f_yg_x-f_xg_y$. We define the tree-model, $T(f,g)$, for the pair, using the contact orders of the Newton-Puiseux roots of $f$ and $g$. We then describe how the $γ$'s climb, and where they leave, the tree. We shall also show by two examples that the way the $γ$'s leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one …
We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are several applications. We discuss one in particular: the calculation of the projection in the tautological ring of the moduli space of abelian varieties of the class of the locus of Jacobians.
Let f_1,...,f_r be homogeneous polynomials in K[x_1,...,x_n], K a field. Put F=y_1f_1+...+y_rf_r in K[x,y] and let I be the ideal of K[x,y] generated by the partials of F relative to the x_i and y_j. The Jacobian ring of F is the quotient J:=K[x,y]/I. We describe J by computing the cohomology of a certain complex whose top cohomology group is J.
$I$-Maximal Variation of Hodge Structure and Jacobian Rings
本文研究光滑超曲面和完全交集家族的高阶Hodge结构变分，通过I-最大变分概念，利用Griffiths的原始上同调描述，将微分Hodge结构变分和n重Yukawa耦合解释为雅可比环的分层乘法映射，证明雅可比环的强Lefschetz性质确保I-最大变分。
For given polynomial map $F:\C^2\to\C^2$ with nonvanishing jacobian we associate a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
We study the intersection cohomology of the theta divisors on Jacobians of nonhyperelliptic curves.
We prove that the cotangent bundle of a complete intersection of two general translates of the theta divisor of the jacobian of a general curve of genus 4 is ample. From this the same result for a general principally polarized abelian variety of dimension 4 follows.
There are two main parts in this manuscript. First, for a Jacobian pair $(f, g)$, with the concept of final major roots and final minor roots, we obtain equations and inequalities for intersection numbers ${\rm I}(f_ξ, g)$ and ${\rm I}(f_ξ, f_y)$ respectively, here, $ξ$ is a generic element of the base field and $f_ξ=f-ξ$ and discuss usage of them in some special cases. Second, we discuss all possibilities of the splits of principle minor roots for the case of degree (99, 66) with help of Abhyankar-Moh planar semigroup, find an unknown possible split and suggest case (99, 66) is open.
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.
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for o…
We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos' normalization algorithm. In the process, we give a simplified proof of the Kunz-Ruppert criterion.
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.
Let C \in P2 be a reduced, singular curve of degree d and equation f = 0. Let Σdenote the jacobian subscheme of C. We have 0 -> E -> 3.O -> I_Σ(d-1) -> 0 (the surjection is given by the partials of f). We study the relationships between the Betti numbers of the module H^0_*(E) and the integers, d; τ, where τ= deg(Σ). We observe that our results apply to any quasi-complete intersection of type (s; s; s).
Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure.
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.
A Counterexample to the Global Injectivity of a Jacobian Mapping in \mathbb{C}^5 and Its Analytical Roots
研究由特定6次齐次多项式诱导的\(\mathbb{C}^5\)到\(\mathbb{C}^5\)的多项式映射\(G\)，证明其雅可比矩阵幂幺但映射非全局单射，构造出\(G(a)=G(b)=\vec{0}\)的点对，还对梯度场零集结构分类得到37个解析复解。
On Strong Lefschetz Property of 0-dimensional Complete Intersections and Associated Forms
研究零维完全交集满足强Lefschetz性质的条件，证明其关联形式的Hessian非零，探讨关联形式的重建方法及其在Jacobian理想中的应用。
Superelliptic jacobians and central simple representations
本文在多项式不可约、Galois 群双传递且极大子群指数满足条件时，证明超椭圆曲线 y^p=f(x) 的 Jacobian 自同态环为第 p 个分圆环。
Let $(P, Q)$ be a pair of Jacobian polynomials. We can show that $ <P, Q>+l+2g(P)-2= 0= <P, [P,Q]>$, where $<f, g>$ is the intersection number of $f, g\in \CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and $g(P)$ is the geometric genus of affine curve defined by $P$. Hence we can show that every Jacobian polynomial defines a smooth rational curve with one point at infinity. It is sufficient to fix the Jacobian conjecture in two dimension by the Abhyankar theorem or the Abhyankar-Moh-Suzuki theorem.
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is …
For a germ $f$ on a complex manifold $X$, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the $D$-module generated by $f^s$ to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of $f^s$ is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane…
We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for line arrangements, (ii) a generalization of Faenzi-Vallès' theorem over a field of arbitrary characteristic, (iii) a partial result on the conjecture of Terao of line arrangements, and (iv) a new sufficient condition for freeness over finite fields. Also, a higher dimensional version of main results is considered.
Let $f(x)$ be a degree $(2g+1)$ monic polynomial with coefficients in an algebraically closed field $K$ with $char(K)\ne 2$ and without repeated roots. Let $\mathfrak{R}\subset K$ be the $(2g+1)$-element set of roots of $f(x)$. Let $\mathcal{C}: y^2=f(x)$ be an odd degree genus $g$ hyperelliptic curve over $K$. Let $J$ be the jacobian of $\mathcal{C}$ and $J[2]\subset J(K)$ the (sub)group of its points of order dividing $2$. We identify $\mathcal{C}$ with the image of its canonical embedding into $J$ (the infinite point of $\mathcal{C}$ goes to the identity element of $J$). Let $P=(a,b)\in \ma…
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous ove…
Covers of curves, Ceresa cycles, and Unlikely intersections
研究通过将Ceresa环与雅可比群中的点关联，证明了非常一般的分叉覆盖曲线的Ceresa环在雅可比群中非 torsion，并给出了具体例子。
We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian. Using methods of additive combinatorics, we are able to give a negative answer to a related question of Chai and Oort where the ambient Shimura Variety is a power of the modular curve.
We present a structure theorem for the multiple non-cyclotomic irreducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents. Roughly speaking, this result shows that the multiple non-cyclotomic irreducible factors of a sparse polynomial, are also sparse. To prove this, we give a variant of a theorem of Bombieri and Zannier on the intersection of a fixed subvariety of codimension 2 of the multiplicative group with all the torsion curves, with bounds having an explicit dependence on the height of the subvariety. We also use this…
New unlikely intersections on elliptic surfaces
本文将复数域上关于 Jacobi 椭圆曲面不可期交集的相切结果扩展至正特征 $p$，并提出了一种研究代数曲线相切性质的新方法，发现了与 Manin 同态相关的新交集族。
On the origin of the Jacobian conjecture
该研究指出雅可比猜想的起源可追溯至1884年克劳斯的论文，尽管其证明存在缺陷，但其思想预示了后续解决方法的发展。
We give a close formula for the Néron-Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number $\hatω^2$ of the dualizing sheaf of a curve in terms of Zhang's invariant $φ$. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the cu…
Koszul complexes and spectra of projective hypersurfaces with isolated singularities
该研究推广了非奇异射影超曲面的相关经典结论，利用Koszul上同调与极点阶谱序列，推导了孤立奇点加权齐次射影超曲面的BS多项式根与Jacobian环挠部的关系。
We give a simplified approach to the Abhyankar--Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.
Root bounds of vertical systems using tropical geometry
本文利用热带几何研究垂直参数化系统的根界，证明了其复零点数与热带线性空间与经典线性空间的交数关系，并在特定情况下用混合体积表示，同时给出正零点数的上界和下界。
Intersection cohomology of type-A toric varieties
本文研究A型环面簇的交点同调，通过GIT商构造，提出组合模型计算Betti数，并证明交点同调群具有典范乘积结构。
Non-isogenous superelliptic jacobians II
本文研究超椭圆曲线的雅可比簇等源性问题，证明当两个多项式一个不可约一个可约时，若对应雅可比簇在代数闭包上等源，则两个自同态代数均含乘法阶为n的可逆元素。
A nine-line counterexample to a conjecture on the minimal degree of Jacobian relations
构造两个具有同构交格但雅可比关系最小度不同的九线排列，给出广义Terao猜想的反例。
Random planar trees and the Jacobian conjecture
本文通过概率方法和随机洗牌子树的理论，提出了雅可比猜想的近似证明，并展示了其在渐近意义上的成立。
Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers
本文通过联络理论中的一般剩余方法研究超平面排列的雅可比理想及其余维数2分支，证明了在温和条件下希尔伯特函数和分次贝蒂数由交集格决定，并给出线排列的新自由性判据。
We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel-Venkatesh mixing conjecture about the equidistribution of CM points…
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a si…
Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may b…
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