Non-proper value set and the Jacobian condition

Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves

We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.

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。

Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic

We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem.

半阿贝尔雅可比簇的Chai猜想

Chai's conjecture for semiabelian Jacobians

本文证明Chai关于半阿贝尔簇基变换导子可加性的猜想在正常曲线雅可比簇情形成立，推广了Raynaud构造，并证明了完美剩余域下Néron模型存在性的两个猜想。

被 Jacobian Kummer 曲面覆盖的 Enriques 曲面上的 Brill-Noether 轨迹

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。

Some remarks on the Jacobian conjecture and polynomial endomorphisms

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of deg…

A note on the Jacobian Conjecture

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.

Pencil of irreducible rational curves and Plane Jacobian conjecture

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.

Analogs of Jacobian conditions for subrings

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of irreducible and square-free elements of the subalgebra k[f1,...,fm]. We also discuss obtained properties in a more general setting - for subrings of unique factorization domains.

Lectures on Geometry and Topology of Polynomials - Surrounding the Jacobian Conjecture

Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one unsurmountable problem remained even in the case of two variables, which has been unsolved since 1939, that is the Jacobian Conjecture. These are notes for author's lectures on the geometry and topology of polynomials and the Jacobian Conjecture delivered at the l'Université de Bordeaux I and at Osaka University in 2003.

On a weak Jelonek's real Jacobian Conjecture in $\R^n$

Let $Y:\R^n\to\R^n$ be a polynomial local diffeomorphism and let $S_Y$ denote the set of not proper points of $Y$. The Jelonek's real Jacobian Conjecture states that if $\codim(S_Y)\geq2$, then $Y$ is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for bijectivity. Furthermore, we generalize our result on bijectivity to semialgebraic local diffeomorphisms.

Distinguished models of intermediate Jacobians

We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also rec…

Autoduality of the compactified Jacobian

We prove the following autoduality theorem for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map A_L: C->J, which maps C into its compactified Jacobian, and form its pullback map A_L^*: Pic^0_J to J, which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then A_L^* is an isomorphism, and forming it commutes with specializing C. Much of our work is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. I…

The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves

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…

The Jacobian Conjecture fails for pseudo-planes

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq …

Test polynomials, retracts, and the Jacobian conjecture

Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever ϕ(p)=p for a mapping ϕof K[x,y], this ϕmust be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping ϕof C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant poly…

Canonical extensions of Néron models of Jacobians

Let A be the Néron model of an abelian variety A_K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-Messing, there is a functorial way to prolong the universal extension of A_K by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. In this paper, we study the canonical extension when A_K=J_K is the Jacobian of a smooth proper and geometrically connected curve X_K over K. Assuming that X_K admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity compo…