Local Tropical Variety
Naoyuki Touda
理想的热带变种是由D.施派尔和B.斯特姆费尔斯提出的。本文在形式幂级数环中给出了局部热带簇的一个类似定义，并证明了类似定理。我们还将讨论当地Groebner风扇和当地热带品种之间的关系。
The tropical variety of an ideal was introduced by D.Speyer and B.Sturmfels. In this paper, we will give an analogous definition of local tropical varieties and prove analogous theorems in the formal power series ring. We will also discuss about relations between the local Groebner fan and local tropical variety.
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We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization $A^\perp$ and the double tropical orthogonalization $A^{\perp \perp}$ of a subset $A$ of the vector space $({\mathbb R} \cup \{ \infty \})^n$. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.
The Gröbner stratification of a tropical variety
热带变种的每个Gröbner层都是一个连通集，所有这些点都诱导出相同的初始子模式。Gröbner分层是Gröbner多面体分解的粗化，其优点是它不依赖于紧化的选择。我们给出了一个例子的曲线在一个领域与非平凡赋值的Gröbner分层严格精细比粗多面体分解的热带品种。我们还表明，Gröbner分层的一个本地拟阵热带品种是完全由底层的热带品种。
Positivity of tropical multidegrees
本研究定义了热带簇的多重次数，证明其正性由热带簇在合适投影映射下的像的维数决定，并以此给出多重射影空间闭子概形多重次数正性判别准则的热带证明。
In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety. Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of…
Tropical expansions and toric variety bundles
本文研究热带展开的几何结构，提出多面体标准判定其为棱锥簇包，并通过例子展示其在一般情况下的极限性。
In this paper we investigate the idea of a tropical critical point of the superpotential for the full flag variety of type A. Recall that associated to an irreducible representation of G=SLn(C) are various polytopes whose integral points parameterize a basis for the representation, e.g. the Gelfand-Zetlin polytope. Such polytopes can be constructed via the theory of geometric crystals by tropicalising a certain function, and in fact, the function involved coincides with the superpotential from the Landau-Ginzburg model for G/B coming from mirror symmetry. In mirror symmetry a special role is p…
Tropical Severi Varieties
我们研究了塞维里品种的热带化，我们称之为热带塞维里品种。在本文中，我们给出了一个部分回答以下问题，“明确描述热带塞维里品种。我们得到一个描述的热带塞维品种在定期细分的多边形。作为中间步骤，我们构造曲线的显式参数空间。这些参数空间是简单得多的对象比相应的塞维品种，他们是密切相关的平坦退化的塞维品种，这反过来又描述了热带塞维品种。作为应用，我们用热带相交理论理解了G. Mikhalkin的对应定理.特别是，这提供了一个证明的独立性的点配置在枚举的热带节曲线。
On Tropical Compactifications
在特征为0的情形下，我们肯定地回答了Tevelev提出的一个猜想：任何簇都包含一个Schön非常仿射开子簇.此外，我们表明，任何球迷支持的热带化的一个schön非常仿射品种产生schön紧化。在离散赋值环上使用复曲面格式，我们将热带相容性推广到非常系数情形。
We construct a family of compactifications of the affine cone of the Grassmannian variety of 2-planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton-Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.
A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.
Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family.
We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. We ask for similar criteria in the realm of tropical linear algebra, giving conditions for two tropical matrices that are polytropes to commute. From the algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimension 2 and 3.
Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in $X$, the associated tropical curves in $S$ deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions…
The Cayley-Menger variety is the Zariski closure of the set of vectors specifying the pairwise squared distances between $n$ points in $\mathbb{R}^d$. This variety is fundamental to algebraic approaches in rigidity theory. We study the tropicalization of the Cayley-Menger variety. In particular, when $d = 2$, we show that it is the Minkowski sum of the set of ultrametrics on $n$ leaves with itself, and we describe its polyhedral structure. We then give a new, tropical, proof of Laman's theorem.
Tropical cycles and Chow polytopes
环面中代数圈的Chow多面体只依赖于它的热带化。概括这一点，我们将周多面体与任何抽象的热带品种在热带复曲面品种。几个重要的多面体与热带品种是特殊情况下，我们的周多面体。热带簇$X$的Chow多面体由一个简单的组合构造给出：它的正规剖分是$X$的Minkowski和和周围环面簇的扇形的反射骨架.
An algorithm for lifting points in a tropical variety
本文的目的是给出热带几何的一个基本定理的构造性证明：给定一个热带簇（用初始理想定义）上的一个点，在代数簇中存在这个点的一个Pu_x值“提升”。这个定理是如此的基本，因为它证明了为什么热带簇（使用初始理想组合定义）携带关于代数簇的信息：它是在赋值映射下的Puixix级数上的代数簇的图像。如果基域是有理数，我们已经使用Singular和Gfan实现了“提升算法”。作为副产品，我们得到了一个算法来计算（K^{n+1}，0）中的空间曲线奇点的Puironx展开。
Totally Nonnegative Tropical Flags and the Totally Nonnegative Flag Dressian
研究完全旗簇及其热带化的全非负部分，证明Lusztig的非负完全旗簇等价于Plücker坐标全非负的旗，并证明完全旗Dressian与热带完全旗簇的全非负部分一致。
The Tropical Variety of Symmetric Rank 2 Matrices
研究对称秩2矩阵簇的热带化，证明其具有对称双色树空间的可剥壳单纯复形结构，讨论相关拟阵结构并给出对称双色树数量的生成函数。
The trigonal construction and the second moment of the tropical Prym variety
本文利用热带三角构造计算亏格不超过4的热带曲线双重覆盖的Prym簇第二矩，结果由多项式与分段多项式项组成，并将后者与Prym--Torelli映射的扩展问题相关联。
For a divisor $D$ on a tropical variety $X$, we define two amounts in order to estimate the value of $h^{0}(X,D)$, which are described by terms of global sections and computed more easily than $h^{0}(X,D)$. As an application of its estimation, we show that a Riemann-Roch inequality holds for smooth tropical toric surfaces.
We define a tropical version $\F_d(\trop X)$ of the Fano Scheme $\F_d(X)$ of a projective variety $X\subseteq \mathbb P^n$ and prove that $\F_d(\trop X)$ is the support of a polyhedral complex contained in $\trop \Grp(d,n)$. In general $\trop \F_d(X)\subseteq \F_d(\trop X)$ but we construct linear spaces $L$ such that $\trop \F_1(X)\subsetneq \F_1(\trop X)$ and show that for a toric variety $\trop \F_d(X)=\F_d(\trop X)$.
Tropical cluster varieties, phylogenetic trees, and generalized associahedra
本文通过将类型C簇变种热带化与轴对称系统发生树空间关联，探讨其有符号热带化作为associahedra或cyclohedra对偶子扇的性质。
A tropical formula for non-Archimedean local heights
本文提出了一种基于热带几何的非阿基米德局部高度计算方法，通过定义delta形式和星积来实现。
Algebraic Properties of Generic Tropical Varieties
证明了在常系数情形下，多项式环中分次理想的代数不变式重数和深度与其泛热带簇的扇形结构密切相关.一般的多重性的理想示出直接对应于一个自然的定义的多样性锥的热带品种。此外，我们可以恢复信息的深度理想的一般热带品种的扇形结构，如果深度是已知的大于0。特别是，在这种情况下，我们可以从它的一般热带变种中看出理想是科恩-麦考利还是几乎科恩-麦考利。
A matroidal perspective on the tropical Prym variety
本文从拟阵视角研究热带Prym簇，将调和双覆盖关联到Zaslavsky符号图拟阵的几何解释，证明Prym簇可由该拟阵加装饰重构，且简化下不变。
Tropical varieties for non-archimedean analytic spaces
我们把环面的解析子簇与热带簇联系起来。在第一部分中，我们将热带代数几何的结果推广到非阿基米德解析情形。周期情形应用于全退化阿贝尔簇A.本文利用严格半稳定模型Y的奇点，得到了从光滑簇Y到A的代数态射值域的维数界。主要结果是，Chambert-Loir的典型措施的封闭子品种A诱导分段Haar措施相关的热带品种，我们给出了明确的描述，这些措施的热带几何。在随后的文件中，这是关键的一步，证明博戈莫洛夫猜想完全退化阿贝尔簇的功能领域。
Generic Tropical Varieties
我们证明了在常系数情形下，分次理想的泛热带簇是存在的。这可以看作是Groebner基理论中一般初始理想存在性的类比。我们确定的一般热带品种作为一组一般和作为一个风扇的主理想和线性理想。
Given a projective variety $X$ of codimension $k+1$ in $\mathbb{P}^n$ the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Tro…
This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Se…
Varieties of prime tropical ideals and the dimension of the coordinate semiring
本文研究了热带多项式半环中素理想变体的性质，并探讨了热带变体与坐标半环维度之间的关系。
Tropical Vector Bundles
本文在热带簇上构造向量丛，特别研究热带环面簇情形，用赋值拟阵及其平坦格描述热带化环面向量丛，并建立整体截面、稳定性及相关滤过理论。
We consider the tropicalization of tangent lines to a complete intersection curve $X$ in $\mathbb{P}^n$. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of $X$ in terms of the tropicalizations of the hypersurfaces cutting out $X$. We apply this to obtain descriptions of the tropicalization of the dual variety $X^*$ and tangential variety $τ(X)$ of $X$. In particular, we are able to compute the degrees of $X^*$ and $τ(X)$ and the Newton polytope of $τ(X)$ without using any elimination theory.
Tropical Implicitization Revisited
本文重新探讨热带隐式化，在Oscar.jl中提出新实现以高效计算超曲面隐式方程的牛顿多面体，并拓展至更高余维及经典隐式化，同时提出开放问题。
Toric degenerations of toric varieties and tropical curves
我们证明了在复曲面素除子的一般位置上的完全复曲面簇上的有理曲线的计数与某些热带曲线的计数是一致的。证明是代数几何的，并依赖于退化技术和日志变形理论。这推广了Mikhalkin的结果，通过不同的方法在表面的情况下，以任意尺寸。
Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford-Taylor theory of plurisubharmonic functions on the tropicalization.
Tropical Geometry over Higher Dimensional Local Fields
本文介绍了定义在高维局部域上的环面的闭子概形的热带化。我们研究这种热带化的基本不变式。这是将Einslieder，Kapranov，Lind，Speyer和Sturmfels的结果推广到更高的局部域。
Computing Tropical Varieties
复系数多项式环中$d$维素理想的热带簇是纯$d$维多面体扇。这个风扇被示出为在余维1中连接。我们提出的算法工具计算的热带品种，我们讨论了我们的实现这些工具中的Gröbner风扇软件\texttt{Gfan}。证明了每个理想都有一个有限的热带基，并给出了一个线性型理想的热带基的大小的下界。
Tropical cluster varieties of type C
本文显式描述了有限C型簇簇的热带化，将其实现为轴对称系统发育树空间，并发现所有坐标符号模式，证明每个符号热带化组合对偶于环面体或结合面体。
Tropical Discriminants
热带几何是用来发展一种新的方法的判别式和结式的理论，在这个意义上的Gel'fand，Kapranov和Zelevinsky。热带A-判别式是由整数矩阵A给出的投射复曲面簇的对偶簇的热带化，它与A的行空间的Minkowski和和A的核的热带化相一致。这导致了一个明确的积极公式的极端单项式的任何A-判别式，没有任何光滑的假设。
On the Tropicalization of the Hilbert Scheme
在这篇文章中，我们研究的热带化的希尔伯特计划和它的适用性作为一个参数空间的热带品种。我们证明了点的热带化的希尔伯特计划有一个热带品种自然与他们有关。为了证明这一点，我们找到了一个约束的程度上的一个理想的热带基地的元素在其希尔伯特多项式。作为推论，我们证明了定义在代数闭值域上的热带簇集只依赖于域的特征对和赋值的象群。总之，我们研究了一些简单的例子，表明热带品种的定义应该包括更多的结构比目前认为的。
In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set, and the second based on the layered structure. We tie them to tropical geometry, especially in connection with the dimension of an affine variety.
We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra $R$ over an algebraically closed field $\mathbf{k}$. Building on work of Römer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain $I$-adic filtrations. As a corollary, we show that in the case that $R$ is a domain, every initial ideal coming from the codimension-$1$ skeleton of the tropical variety is prime, so "generic presentations of Cohen-Macaulay domains are well-poised in codimension-$1$."
Period integrals of hypersurfaces via tropical geometry
本文利用热带几何方法计算环面簇中一参数复超曲面的周期积分渐近，并给出 d=1 时极限处的极化对数 Hodge 结构。
We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type $A$. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.
We first develop the local theory of functions on $\mathbb R^n$ defined by tropical Laurent polynomials. We study the structure of the semiring of functions, where two functions are identified when they coincide on a neighborhood of a fixed point. We see that this semiring is closely related to the semiring of functions defined by Boolean Laurent polynomials. Then we develop the local theory of functions on tropical curves. We construct a contravariant functor from the category of 1-dimensional tropical fans to the category of certain homomorphisms of semirings. As an application, we discuss a…
Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of var…
When are tropical multidegrees positive?
该论文研究热带多重度的正性，引入投影纯性与 facet 可选择性条件，推广 He 的定理，同时指出增广 Bergman 扇对应的热带体积多项式具有洛伦兹型性质。
Generic Tropical Varieties on Subvarieties and in the Non-constant Coefficient Case
在较早的文件中，它表明，一般的热带品种的理想可以包含信息的代数不变式，例如深度在一个直接的方式。到目前为止，在常系数的情况下，对于通常的属的概念，属的热带变种的存在已经被证明。在本文中，我们推广这个存在性结果，包括在一定的设置非常数系数的情况下。此外，我们将一般性的概念推广到一般线性群的任意闭子簇。除了包括概念的通用性代数群这产生结构的结果热带品种的理想下的任意线性坐标的变化。
We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an application, we study nearby monodromy eigenvalues -- the eigenvalues of the monodromy operators of singularities, adjacent to a given singularity of a holomorphic function $f$. More precisely, we express some of such values in terms of certain resolutions of $f$. The expression is given it terms of the exceptional divisor strata of arbitrary codimension, general…
We give a description of the tropical variety of univariate polynomials of degree n having two double roots. As a set, it is given as the union of three types of maximal cones of dimension n-1, where only cones of two of these types are cones of the secondary fan of {0,...,n}. Through Kapranov's theorem, this goal is achieved by a careful study of the possible valuations of the elementary symmetric functions of the roots of a polynomial with two double root. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients.
Jacobian varieties of reduced tropical curves
在$\rr^2$的热带几何中，除数和雅可比簇的定义类似于代数几何。为了研究这些天体，考虑热带曲线的“束”是很重要的。在本文中，我们将证明，如果束是一个花束，那么雅可比矩阵是一个高维环面。
Every tropical ideal in the sense of Maclagan-Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements, and in which all maximal cones have we…
Tropical vector bundles and matroids
本研究提出热带环面簇上的热带向量丛（环面拟阵丛）概念，定义其等变K-理论、示性类等，证明拟阵的泛丛对应关系，建立正性概念与上同调零化结果，并研究向量丛分裂的Grothendieck定理热带版本。
Non-archimedean amoebas and tropical varieties
我们研究了非阿基米德对应复杂的变形虫的代数品种，并表明，它符合一个多面体集定义的Bieri和格罗夫斯使用估值。对于超曲面，这个集合也是定义多项式的热带簇。使用非阿基米德分析和康拉德最近的结果，我们证明了阿米巴的一个不可约的品种是连接。我们引入了一个adelic变形虫的概念，品种在全球领域，并建立了一种形式的本地全球原则。这个原理被用来解释一个相关的动力系统的非扩张集的计算。
Introduction to $PSL_2$ phase tropicalization
本文在矩阵群 $PSL_2(\mathbb{C})$ 上引入相位热带化的非交换版本，将经典代数环面 $(\mathbb{C}^*)^n$ 中的相位热带簇方法推广到非交换情形。
To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a prime ideal. We also show that there are infinitely many Bergman fans that are not set-theoretically realizable as the tropicalization of any ideal.
We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme struct…
Tropical bases by regular projections
我们认为热带品种$\mathcal{T}(I)$的素理想$I$所产生的多项式$f_1, ..., f_r$，并重新定期介绍的投影技术，由Bieri和Groves从计算的角度来看。特别是，我们表明，$I$有一个短期的热带基数最多$r + \codim I + 1$的价格增加度，我们提供了一个计算描述这些基地。
The Tropical Grassmannian
在热带代数几何中，多项式方程的解集是分段线性的。我们介绍了热带品种的多项式理想，我们确定它与多面体子复杂的Grobner风扇。热带的格拉斯曼方程就是以这种方式从二次普拉克关系的理想中产生的。它被证明参数化所有热带线性空间。热带射影空间中的线是树，其热带格拉斯曼G_{2,n}等于Billera，Holmes和Vogtmann研究的系统发育树的空间。高等格拉斯曼学派提供了树的空间的自然概括。他们的方面对应于二项式初始理想的Plucker理想。热带格拉斯曼型G_{3,6}是由1035个四面体胶合而成的单纯复形。
The Vámos Matroid Has No Second Symmetric Power
本文证明 Vámos 拟阵 $V_8$ 没有第二对称幂，从而否定地回答了 Anderson 的问题，并由此得出其相关热带线性空间不能由热带理想实现。
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