Lars Allermann, Johannes Rau
We introduce an improved version of rational equivalence in tropical intersection theory which can be seen as a replacement of chapter 8 of our previous article arXiv:0709.3705v2. Using this new definition, rational equivalence is compatible with push-forwards of cycles. Moreover, we prove that every tropical cycle in R^r is equivalent to a uniquely determined affine cycle, called its degree.
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We study moduli spaces of rational graphically stable tropical curves and a refinement given by radial alignment. Given a complete multipartite graph $Γ$, the moduli space of radially aligned $Γ$-stable tropical curves can be given the structure of a balanced fan. This fan structure coincides with the Bergman fan of the cycle matroid of $Γ$.
We compute the $S_n$-equivariant rational homology of the tropical moduli spaces $Δ_{2,n}$ for $n\leq 8$ using a cellular chain complex for symmetric $Δ$-complexes in Sage.
We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces a…
Equality of tropical rank and dimension for semimodules of tropical rational functions, and computational aspects
本文证明了热带有理函数semimodule的热带秩等于其拓扑维数，并探讨了计算热带秩的复杂性。
Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles
研究了有理双Hurwitz圈，即在两个固定分支点上有一个映射到射影直线上的有理稳定曲线的轨迹。推广双Hurwitz数观察到的现象，这样的周期是分段多项式的特殊分歧的条目，多项式和壁交叉室有一个明确的和“模块化”的描述。本文的一个主要目标是同时进行这项调查的相应对象在热带几何，强调一个精确的组合之间的经典和热带赫维茨理论的对偶性。
Tropical Incidence Relations, Polytopes, and Concordant Matroids
在本文中，我们开发了一个热带模拟的经典旗品种，我们称之为标志Dressian。我们发现的关系，我们称之为"热带发病关系"，当一个热带线性空间包含在另一个，并表明，标志Dressian是一个热带prevariety。在2步旗Dressians的情况下，我们称之为“热带发病率prevariants”，我们发现在这个空间中的点之间的等价性和诱导的超单形的细分，推广两个部分的等价D。Speyer关于热带线性空间的一个定理。我们试图将Speyer等价的第三部分推广到协调拟阵，并得到一些部分结果。
Chern classes of tropical vector bundles
引入了热带向量丛、这些丛的态射和有理截，定义了热带向量丛和有理截沿态射的拉回。然后利用热带向量丛的有界有理截面定义了该向量丛的Chern类，并证明了Chern类的一些基本性质。最后，我们给出了椭圆曲线上所有向量丛在同构下的完全分类。
We study moduli spaces of rational weighted stable tropical curves, and their connections with the classical Hassett spaces. Given a vector w of weights, the moduli space of tropical w-stable curves can be given the structure of a balanced fan if and only if w has only heavy and light entries. In this case, we can express the moduli space as the Bergman fan of a graphic matroid. Furthermore, we realize the tropical moduli space as a geometric tropicalization, and as a Berkovich skeleton, of the classical moduli space. This builds on previous work of Tevelev, Gibney--Maclagan, and Abramovich--C…
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio condit…
We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational functions with their degrees. As a corollary, we show that the automorphism group of a tropical curve is isomorphic to the $\boldsymbol{T}$-algebra automorphism group of its rational function semifield, where $\boldsymbol{T} := (\boldsymbol{R} \cup \{ -\infty \}, \operatorname{max}, +)$ is the tropical semifield. Finally, we describe all semiring automorphi…
We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield $\boldsymbol{T} := ( \boldsymbol{R} \cup \{ - \infty \}, \operatorname{max}, +)$ by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves $φ: \varGamma \to \varGamma^{\prime}$, the rational function semifield of $\varGamma$ is finitely generated as a $φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$-algebra, where $φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of the rational…
Intersecting Psi-classes on tropical M_{0,n}
我们应用L. Allermann和J. Rau在有理热带曲线的模空间上计算热带Psi类的交积。证明了在零维（稳定）相交的情况下，所得的数与代数几何中计算的n-标记有理曲线模空间上的Psi类的相交数一致.
Moduli spaces of rational tropical curves
本文定义了n个标志点的有理热带曲线的模空间。我们表明，这个空间有一个结构的光滑热带品种的尺寸n-3。我们定义了这个空间的Deligne-Mumford紧化和热带$ψ$类因子。
Rational curves in Fano hypersurfaces and tropical curves
利用热带曲线和退化理论的思想，我们证明了任何Fano超曲面（更一般的Fano完全相交）是由最多的二次有理曲线扫描。
The Tropical Moduli Space of Degree-3 Rational Maps
本文研究了三次热带有理映射的热带模空间，通过斜率序列的组合描述，分类了所有映射并证明存在恰好十种组合类型，给出了由断裂点间间隙长度参数化的多面体模型。
We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked four-valent vertex. Another refined invariant counts rational tropical curves of a given de…
Universal families of rational tropical curves
我们引入了n-标记光滑有理热带曲线族在光滑热带品种的概念，并建立了一个一一对应（等价类）这些家庭和态射从光滑热带品种到模空间的n-标记抽象有理热带曲线。
Finitely generated congruences on tropical rational function semifields
本文证明了热带有理函数半域上的同余有限生成等价于其对应集合闭包为有限个有理多面体集的并，并据此刻画了热带曲线的有理函数半域。
Tropical descendant Gromov-Witten invariants
我们在R^2中有理热带曲线的模空间上定义了热带Psi-类，并考虑了这个空间上Psi-类的交积和求值的拉回。我们证明了一个WDVV方程，它足以证明满足一定Psi-和评价条件的曲线的热带数等于相应的经典数。我们提出了一种算法，推广Mikhalkin的格路径算法，并计算合理的平面热带曲线满足一定的PSI和评价条件。
$\boldsymbol{T}$-algebra homomorphisms between rational function semifields of tropical curves
本文证明了两个热带曲线有理函数半域间的单射 $\boldsymbol{T}$-代数同态能诱导出这两个热带曲线间的满射态射。
In this paper we introduce a refined multiplicity for rational tropical curves in arbitrary dimension, which generalizes the refined multiplicity introduced by F. Block and L. Göttsche in arXiv:1407.2901 . We then prove an invariance statement for the count of rational tropical curves in several enumerative problems using this new refined multiplicity. This leads to the definition of Block-Göttsche polynomials in any dimension.
For a rational map $ϕ$ from a metric graph $\varGamma$ to a tropical projective space $\boldsymbol{TP^n}$ defined by a ratio of rational functions $f_1, \ldots, f_{n + 1}$, an automorphism $σ$ of $\varGamma$ induces a permutation of the coordinates of $\boldsymbol{TP^n}$ if $\{ f_1, \ldots, f_{n + 1} \}$ is $\langle σ\rangle$-invariant. Through this description, we can realize the automorphism group of $Γ$ as ambient automorphism group such as tropical projective general linear group, tropical general linear group and $\boldsymbol{Z}$-linear transformation group of Euclidean space.
An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are designed for decomposing tropical algebraic rational functions being (in the classical language) piece-wise linear functions with rational slopes of their linear pieces. In addition, we provide a criterion when the composition of two tropical polynomials commutes (for classical polynomials a similar question was answered by J.~Ritt).
Tropical rational curves with first order tangency via WDVV
本文利用 WDVV 技术的热带类比，计算了 $\mathbb{CP}^2$ 中具有一阶切触的热带有理曲线计数，验证了结果与复几何中的早期结果一致。
This article discusses the concept of rational equivalence in tropical geometry (and replaces the older and imperfect version arXiv:0811.2860). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the "bounded" Chow groups of $\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles i…
We study the topology of the link $M^{\mathrm{trop}}_{g,n}[1]$ of the tropical moduli spaces of curves when g=2. Tropical moduli spaces can be identified with boundary complexes for $\mathcal{M}_{g,n}$, as shown by Abramovich-Caporaso-Payne, so their reduced rational homology encodes top-weight rational cohomology of the complex moduli spaces $\mathcal{M}_{g,n}$. We prove that $M^{\mathrm{trop}}_{2,n}[1]$ is an $n$-connected topological space whose reduced integral homology is supported in the top two degrees only. We compute the reduced Euler characteristic of $M^{\mathrm{trop}}_{g,n}[1]$ for…
First Steps in Tropical Intersection Theory
我们建立了热带相交理论的第一部分。也就是说，我们定义了循环，卡地亚因子和这两者之间的交集产品（不通过有理等价），并讨论了推进和拉回。我们首先对R^n中的fans做这个，然后对局部fans的“抽象”循环做这个。关于在计数几何中的应用，我们最后看一下R^n中圈和圈类的有理等价和交积。
We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in $\mathbb P^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.
Topological Classification of Univariate Sparse Polynomials via Tropical Geometry
该研究利用热带几何对支撑集为$A=\{d,d+1,d+2,d+3,d+4\}(d\geq3)$、具两个$(2,2)$型临界值的单变量稀疏复多项式分类，发现其拓扑等价类与无穷远分支热带化组合类型的对应关系及支撑集平移下的可能周期性。
Tropical Quadrics Through Three Points
我们热带化的有理映射，采取三重点的射影平面到平面的二次曲面通过这些点。它的热带化的意象包含在它的意象的热带化之中。我们确定这些物体内的热带格拉斯曼平面投影5-空间，我们探索一个小的热带希尔伯特计划。
We introduce tropical Newton-Puiseux polynomials admitting rational exponents. A resolution of a tropical hypersurface is defined by means of a tropical Newton-Puiseux polynomial. A polynomial complexity algorithm for resolubility of a tropical curve is designed. The complexity of resolubility of tropical prevarieties of arbitrary codimensions is studied. Tropical Newton-Puiseux rational functions are introduced, and we prove that any tropical polynomial has a resolution in tropical Newton-Puiseux rational functions (this can be treated as a tropical analog of the algebraic closedness of the f…
Minimum volumes of tropical rational functions
本文研究了热带有理函数的最小体积表示问题，发展了对偶性定理来定义对 (f, g) 的体积，并证明了一维情况下最小体积表示的存在性与对偶细分的唯一性，而二维情况下唯一性不成立。
A tropical framework for using Porteous formula
本文提出一种热带框架，利用分裂原理证明了Porteous公式在热带几何中的应用，通过研究特征类和退化领域的基本类来建立其理论基础。
Tropical representation of Weyl groups associated with certain rational varieties
从（P^1）^N的某些有理簇出发，我们构造了一个热带的，即，无减双有理，表示外尔群作为一组伪同构的品种。此外，我们开发了一个代数几何框架的τ-函数作为定义功能的例外因子的品种。在相应的根系是仿射型的情况下，我们的结构产生了一类（高阶）q-差分Painleve方程，其代数次数二次增长。
Irreducible cycles and points in special position in moduli spaces for tropical curves
本文第一部分讨论了n-标记有理热带曲线的模空间中圈的不可约性。我们证明了PSI类和重要的因子是不可约的，并且局部不可约因子也是全局不可约的n \leq 6。本文的第二部分证明了（\R^2）^n中的点位形在计数有理平面曲线（两种不同的定义）的特殊位置上的轨迹可以给出余维为1的热带循环的结构。此外，我们显式地计算这个循环的权重。
Matroid Products in Tropical Geometry
本文建立了具有任意大对称幂的赋值拟阵与作为热带理想簇的热带线性空间之间的等价关系，并利用该结果证明具有二阶对称幂的拟阵类是次闭的且具有无穷多个禁用次子。
Congruences on tropical rational function semifields and tropical curves
本文定义了热带有理函数半域，通过具体映射证明热带曲线可作为同余簇实现，并揭示了商半域在决定同构类中发挥类似坐标环的作用。
We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.
We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig for tropical moduli spaces of rational stable curves with distinct marked points. Our computation reveals that the weight of a maximal cone in an intersection has a combinatorial intepretation in terms of the underlying tropical curve and it is always nonnegative. In particular, our result specialises to that, in top dimension, the tropical intersection product coincides with its classical counterpart.
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。
We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new…
Tropical curves with parallel rays
本文针对传统热带曲线无法处理平行射线的问题，提出新的抽象热带曲线定义，并通过范畴等价性将几何概念转化为代数概念。
We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, in…
Tropical fans and the moduli spaces of tropical curves
我们给出了一个严格的定义的热带球迷（“当地积木的热带品种”）和他们的态射。对于这样一个态射的热带风扇的相同尺寸，我们表明，逆像的数量（计算与适当的热带多重性）的一个点在目标上不依赖于所选择的点-一个声明，可以被视为开始的热带相交理论。作为一个应用，我们考虑有理热带曲线（抽象的和在某些R^r）的模空间与评价和遗忘态射。利用我们的结果，这给出了各种热带独立性陈述的新的、简单的和统一的证明，例如，通过给定点的有理热带曲线的数目（在任何R^r中）与点无关的事实。
Automorphism Groups on Tropical Curves: Some Cohomology Calculations
设$X$是抽象热带曲线，$G$是$X$的自同构群的有限子群.设$D$是等价类为$G$ -不变的$X$上的除数.我们解决以下问题：是否总是有一个除数$D'$在等价类的$D$是$G$不变的？我们的主要结果是，答案是“是”的所有抽象的热带曲线。在我们的证明中的一个关键步骤是希尔伯特定理90的热带模拟。
The diagonal of tropical matroid varieties and cycle intersections
我们定义了一个交叉产品的热带循环的拟阵品种（通过削减了对角线），并表明它是良好的行为。特别是，这使我们能够相交的热带有理标记曲线$M_{0,n}$和$M_{0,n}(\R^r, d)$的模空间上的周期。这种交积可以推广到光滑簇（其局部模型是拟阵簇）。我们还研究了回拉循环和有理等价。
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.
Quadratically Enriched Plane Curve Counting via Tropical Geometry
本文通过热带几何中的地板图技术，证明光滑环面del Pezzo曲面中经过有理点和共轭点的有理曲线的二次加丰富计数可由特定多重性的垂直拉伸点条件的热带稳定映射计数确定，并给出计算算法。
Demailly showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents associated to subvarieties, and asked whether any strongly positive (p,p)-dimensional closed current with rational cohomology class can be approximated by positive linear combinations of integration currents associated to subvarieties. Using tropical geometry, we construct a (p,p)-dimensional current on a smooth projective variety that does not satisfy the latter statement.
Tropical intersection homology
本文通过引入热带交义上同调，将代数循环的数值等价推广到合适的光滑正则簇与除子对，结合热带几何与上同调理论。
We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in R^r as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to R^r. The weights of the top-dimensional polyhedra are given in terms of certain lattice indices and local Hurwitz numbers.
To every rational complex curve $C \subset (\mathbf{C}^\times)^n$ we associate a rational tropical curve $Γ\subset \mathbf{R}^n$ so that the amoeba $\mathcal{A}(C) \subset \mathbf{R}^n$ of $C$ is within a bounded distance from $Γ$. In accordance with the terminology introduced by Passare and Rullgård, we call $Γ$ the spine of $\mathcal{A}(C)$. We use spines to describe tropical limits of sequences of rational complex curves.
This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $…
Let X and X' be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles, in the sense of Henning Meyer's graduate thesis, between the tropicalization of the intersection product of X and X' and the stable intersection of trop(X) and trop(X'), when restricted to (the inverse image under the tropicalization map of) a connected component C of the intersection of trop(X) and trop(X'). This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection …
On Bands and Limit Theorems in Tropical Geometry
本文基于Baker-Jin-Lorscheid提出的带理论，研究了带方案在热带化、解析化和F₁几何中的应用，并证明了通过带方案的极限得到Payne极限定理的方案论增强。
Algebraically trivial tropical cycles are smash-nilpotent
该研究证明热带簇中代数平凡的热带闭链是smash-幂零的，通过构造有理等价完成证明，还提出Voevodsky猜想的热带版本并探讨其在辛几何中的应用。
The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves
设K是关于非平凡的非阿基米德赋值完备的代数闭域，Λ是它的值群.给定一个光滑的、适当的、连通的K曲线X和布氏分析X^ \an的骨架Γ，有两个自然的实环面可以考虑：热带雅可比Jac（Γ）和布氏分析Jac（X）^ \an的骨架。我们表明，雅可比矩阵的骨架是规范同构的雅可比矩阵的骨架，主要是极化热带阿贝尔簇。此外，我们还证明了经典Abel-Jacobi映射的热带化是热带Abel-Jacobi映射。作为这些结果的一个结果，我们推出了在热带几何意义下，Γ上的Λ-有理主除子正是X上的主除子的收缩.我们实际上证明了一个更精确的结果，即虽然因子的零点和极点在收缩映射下可以抵消，但为了将Γ上的一个Λ-有理主除子提升为X上的一个主除子，不需要增加超过g个额外的零点和g个额外的极点。我们的结果意味着：对于X上的某个非零亚纯函数f，连续函数F：Γ-> R是对-log |f|的Γ的限制当且仅当F是一个Λ-有理热带亚纯函数，并且我们利用这一事实证明了存在一个有理映射f：X --> P^3，当其热带化被限制在Γ上时，它在其象上是等距的.
We introduce a \textit{quantum index} for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2-form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the results from G. Mikhalkin arXiv:1505.04338 to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.
Tropical singular intersection homology
本文引入了带有热带系数的热带奇异交同调（非GM和GM），研究其基本性质，并在某些情况下计算了同调。
Tropical descendant invariants with line constraints
本文针对带线约束和Psi-条件的有理下降对数Gromov--Witten不变量，提出基于热带几何的递归计算方法，并推广到两条线的情形。
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