Grigory Mikhalkin, Ilia Zharkov
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
On tropical geometry in $\rr^2$, the divisor and the Jacobian variety are defined in analogy to algebraic geometry. For study of these objects, it is important to think of the `bunch' of a tropical curve. In this paper, we will show that if the bunch is a bouquet, then the Jacobian is a higher-dimensional torus.
By tropical Abel-Jacobi theorem, the Jacobian of a tropical curve is isomorphic to the Picard group. A tropical curve in $\mathbb{R}^2$ corresponds to an immersion from a tropical curve to $\mathbb{R}^2$. In this paper, we show that any principal divisor on a tropical curve is the restriction of a principal divisor on the ambient plane $\mathbb{R}^2$.
We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus $g$. This space parametrizes equivalence classes of tropical curves of genus $g$ together with a $μ$-polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decomposit…
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, the tau invariant, and the total length of a metric graph. We argue that this linear relation is a non-archimedean analogue of a recent remarkable ide…
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the resulting weighted metric graph. In general, it is not known how to find the abstract tropicalization of a curve defined by polynomial equations, since an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction in practice. We solve this problem in the case of hyperelliptic curves by studying admissible co…
Tautological cycles on tropical Jacobians
该研究证明了经典庞加莱公式的热带版本，同时得到带整结构实环面与热带阿贝尔簇的若干基础结果（包括热带Appell-Humbert定理），并探究了热带闭链的不同等价概念及其关联。
A $K_4$-curve is a smooth, proper curve X of genus 3 over a nonarchimedean field whose Berkovich skeleton $Γ$ is a complete graph on 4 vertices. The curve X has 28 effective theta characteristics, i.e. the 28 bitangents to a canonical embedding, while $Γ$ has exactly seven tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a $K_4$-curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.
This note is a follow up of math.AG/0612267v2 and it is largely inspired by a beautiful description of Baker-Norine of non-effective degree (g-1) divisors via chip-firing game. We consider the set of all theta characteristics on a tropical curve and identify the Riemann constant as a unique non-effective one among them.
Tropical split Jacobians of curves of genus 2 II
本文研究亏格2曲线的热带分裂雅可比簇，通过分析一对椭圆曲线及其乘积的有限子群这一构建块，探讨如何将其重新组装为雅可比簇，区别于前作的整体性视角。
We give an explicit description of theta-characteristics on tropical curves and characterize the effective ones. We construct the moduli space for tropical theta-characteristics of genus g as a generalized cone complex. We describe the fibers of the specialization map from the classical moduli space of theta-characteristics on algebraic curves to our moduli space of tropical theta-characteristics.
Tropical manifolds are polyhedral complexes enhanced with certain kind of affine structure. This structure manifests itself through a particular cohomology class which we call the eigenwave of a tropical manifold. Other wave classes of similar type are responsible for deformations of the tropical structure. If a tropical manifold is approximable by a 1-parametric family of complex manifolds then the eigenwave records the monodromy of the family around the tropical limit. With the help of tropical homology and the eigenwave we define tropical intermediate Jacobians which can be viewed as tropic…
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.
For each universal genus-$g$ polarization $μ$ of degree $d$, we construct a universal tropical Jacobian $J_{μ,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{μ,g}^{trop}$. In particular, we prove that the natural compactification of $J_{μ,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.
We show that the Abel-Jacobi image of a tropical curve $C$ in its Jacobian $J(C)$ is not algebraically equivalent to its reflection by using a simple calculation in tropical homology.
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.
Tropical Split Jacobians of genus 2 and optimal covers
本文探讨了热带阿贝尔簇与热带曲线范畴的联系，通过研究热带分裂雅可比簇现象，证明了最优覆盖能解决亏格2曲线雅可比簇分裂的非唯一性与计算难题，将抽象机制与算法方法相结合。
Tropicalization of the universal Jacobian
本文提供了栈理论框架研究热带曲线模空间上的泛热带 Jacobian，从对数与非阿基米德解析角度证明泛紧化 Jacobian 的热带化即为泛热带 Jacobian，且热带化映射与重言态射相容。
Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture o…
We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--Lütkebohmert theory of non-Archimedean theta functions for…
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.
We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the surjection is reduced to the bijection on an appropriate subset of the set of effective divisors of degree $g$ on the curve. Thus the subset of effective divisors has the additive group structure induced from the Jacobian. We finally realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ via the intersection with a tropical curve of degre…
This is an expository paper about the topics listed in the title.
We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical…
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…
In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.
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.
We study the intersection cohomology of the theta divisors on Jacobians of nonhyperelliptic curves.
We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described by Gross, Hacking, and Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, M…
We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends only on the underlying graph of a tropical curve and is preserved when passing to genus $\geq 2$ connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.
Effective faithful tropicalizations and embeddings for abelian varieties
本文通过非阿基米德theta函数实现阿贝尔簇的热带化，提供经典勒夫定理的热带版本，结合热带theta函数嵌入与非阿基米德theta函数提升，拓展了热带几何的应用范围。
We prove that k-th singular cohomology group of the complement of the theta divisor in a hyperelliptic Jacobian is isomorphic to the k-th fundamental representation of the symplectic group Sp(2g,C). This is one of the conjectures in our previous paper.
Tropical Abel-Jacobi theory
该研究针对任意维紧热带簇，通过热带同调与单值化定义中间Jacobian并构建函子性Abel-Jacobi映射以发展热带Abel-Jacobi理论，恢复1维曲线理论并给出热带曲线Ceresa类的组合显式公式。
We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.
We show that the only summands of theta divisors on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the obvious powers of the curve and the Fano surface of lines on the threefold. The proof uses the decomposition theorem for perverse sheaves, a bit of representation theory and a computation of characteristic cycles for Brill-Noether sheaves.
Chip-firing games, Jacobians, and Prym varieties
该研究对度量图芯片射击游戏理论与热带Prym簇理论做自成体系的介绍，探讨二者和代数对应概念的关联，提出未来研究方向。
We show how to calculate the Euler characteristic of an affine Jacobi variety of a spectral curve from its defining equations.
We discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when such a curve is real. We then revisit a curve constructed by Emch with the greatest known number of real tritangents, and conversely construct a curve with very few real tritangents. Using recent results on the relation between algebraic and tropical theta characteristics, we show that the tropicalization of a canonical sextic curve has 15 tritangent planes.
In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the…
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne-Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
The Jacobian $J$ of a complete, smooth, connected curve $X$ admits a canonical divisor $Θ$, called the Theta divisor. It is well-known that $Θ$ is ample and, in fact, $3Θ$ is very ample. For a general complete, integral curve $X$, D'Souza constructed a compactification $\bar J$ of the Jacobian $J$ by considering torsion-free, rank 1 sheaves on $X$. Soucaris and the author considered independently the analogous Theta divisor $Θ$ on $\bar J$, and showed that $Θ$ is ample. In this article, we show that $nΘ$ is very ample for $n$ greater or equal to a specified lower bound. If $X$ has at most ordi…
We propose a method to study the integrable cellular automata with periodic boundary conditions, via the tropical spectral curve and its Jacobian. We introduce the tropical version of eigenvector map from the isolevel set to a divisor class on the tropical hyperelliptic curve. We also provide some conjectures related to the divisor class and the Jacobian. Finally, we apply our method to the periodic box and ball system and clarify the algebro-geometrical meaning of the real torus introduced for its initial value problem.
The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently be…
Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for X. This result is useful for reducing certain questions about the Abel-Jacobi map Phi_p : X -> J(X), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over the rationals. As an application of…
We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Carta…
Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Fina…
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2Θ$. The embedded tangent space at a semi-stable non-stable bundle $ξ\oplusξ^{-1}$, where $ξ$ is a degree zero line bundle, is shown to consist of those divisors in $|2Θ|$ which contain $Sing(Θ_ξ)$ where $Θ_ξ$ is the translate of $Θ$ by $ξ$. We also obtain geometrical results on the structure of this tangent space.
Some Differential Equations for the Riemann $θ$-Function on Jacobians
本文利用受Arakelov理论启发的Fay公式变体，证明了黎曼曲面雅可比簇上黎曼θ函数满足的若干微分方程。
We give a criterion to distinguish between a genus three Jacobian and its [-1] twist in terms of the product of the 36 even theta nulls. We also express the product of the 36 theta nulls in terms of the discriminant of a genus three curve. The results are arithmetic in nature and thus add to previous work over C on the product of the even theta nulls. They generalize previous work of Ritzenthaler and Lachaud for Abelian threefolds which are (2,2,2) isogenous to a product of elliptic curves. This answers a 2003 of letter of Jean-Pierre Serre to Jaap Top.
We reconcile the discrepancy between the complex and tropical counts of some enumerative problems reducing to positive characteristic. Each problem that we consider suggests a prime with special behaviour. Modulo this prime, the solutions coalesce in uniform clusters: at this special prime, the geometric and the tropical behaviours match. As examples, we concentrate on inflection points of plane curves and theta-hyperplanes of canonical curves.
Combinatorics and real lifts of bitangents to tropical quartic curves
本文研究热带平面四次曲线双切线的七个等价类，给出其41种类型的完整组合分类，并建立显式符号规则以构造实提升，证实每类恰有零个或四个全实提升。
We prove the following converse of Riemann's Theorem: let (A,Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ=C+Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.
The tropical $n$-gonal construction
本研究提出Donagi n边形构造的纯热带类似物，利用热带同调理论分析其组合性质，证明n=2、3情形下的对偶性、可逆性及簇同构关系，证明思路遵循代数版本。
We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding addition map, and show that the minimum can only be achieved if X is a theta divisor. Conjecturally, the latter happens only on Jacobians of curves and intermediate Jacobians of cubic threefolds. As an application, we prove that nondegenerate generic vanishing subschemes of indecomposable principally polarized abelian varieties are automatically reduced and irreduc…
Theta functions for singular curves
本文研究了奇异黎曼曲面的θ函数，通过构造通用雅可比簇的紧化，扩展了黎曼的经典结果。
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my talks at the 2015 AMS symposium in algebraic geometry and at AGNES 2016, is to present some of the results in this area.
Tropical KP Theory on Banana Curves
本文研究了香蕉曲线与热带KP理论的关系，通过组合学方法将热带theta除子与matroid和Grassmannian结构联系起来，并定义了Hirota流形以参数化tau函数。
We show that the degree of the images of the moduli space of (principally polarized) abelian varieties A_g and of the moduli space of curves M_g in the projective space under the theta constant embedding are equal to the top self-intersection numbers of one half the first Hodge class on them. This allows us to obtain an explicit formula for the degree of A_g, and an explicit upper bound for the degree of M_g. Knowing the degree of A_g allows us to effectively determine the subvariety itself, i.e. to effectively obtain all polynomial equations satisfied by theta constants. Furthermore, combinin…
Tropical Matsusaka and Matsusaka--Ran Criteria
本文证明热带松阪准则与热带松阪-兰准则，结合热带庞加莱公式为热带肖特基问题提供几何解，还开发了相关工具并回答了Brannetti等人的问题。
We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。