Andreas Gathmann
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction to these new techniques with a special emphasis on the recent applications to problems in enumerative geometry.
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Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo surfaces.
Tropical and algebraic elliptic plane curves with fixed j-invariant
该研究通过热带计数特定椭圆曲线，结合对应定理，给出代数计数的另一种证明，拓展了枚举几何的相关方法。
In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.
We show that the non-Archimedean skeleton of the $d$-th symmetric power of a smooth projective algebraic curve $X$ is naturally isomorphic to the $d$-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of $X$. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under…
We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to the moduli spaces of smooth and stable algebraic curves, from the combinatorial, the topological, and the Teichmüller point of view. The paper is written in an expository style, and it generalizes some results contained in sections 4-6 of arXiv:1001.2815v3.
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.
We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.
This is mostly* a non-technical exposition of the joint work arXiv:1212.0373 with Caporaso and Payne. Topics include: Moduli of Riemann surfaces / algebraic curves; Deligne-Mumford compactification; Dual graphs and the combinatorics of the compactification; Tropical curves and their moduli; Non-archimedean geometry and comparison. * Maybe the last section is technical.
Realization of tropical curves in abelian surfaces
利用实环面中的热带曲线构造阿贝尔曲面中的代数曲线，给出热带曲线可实现的条件，并推导组合计数公式，建立类似Mikhalkin对应定理的代数-热带对应。
Genus one correspondence between tropical and algebraic curves
研究热带曲线与代数曲线的亏格一对应，通过对数形变理论及固定热带化的对数映射枚举，证明代数椭圆曲线计数与热带曲线计数一致，推广了亏格0的对应定理。
We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization is a finite union of polyhedra. Previously, the converse direction was known by the theorem of Bieri and Groves. Over the field of complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for analytic subvarieties of tori.
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach in classical algebraic geometry. The bend loci of an ideal over tropical dual numbers coincides with the bend loci of a congruence over tropical semiring and this enables tropical dual numbers to serve as an algebraic structure for tropical geometry. Tropical Zariski topology on affine tropical spaces whose closed sets are non-linear loci of ideals over tro…
On tropical cohomology of smooth algebraic varieties
本文显式描述了平凡赋值域上光滑代数簇的热带上同调，并为几何严格半稳定约化构造了单值权重谱序列。
Tropical Tevelev degrees
本文定义热带Tevelev次数，通过构造计算其为2^g，并证明其与代数次数一致。
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it computes sufficient conditions to have an algebraic counterpart related by tropicalization. We also provide sufficient conditions in a geometric construction to ensure that the algebraic counterpart always exists. Geometric constructions are applied to transfer classical theorems to the tropical framework, we provide a notion of incidence theorems and prove sev…
Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition reproduces a generalization of Speyer's well-spacedness condition and also gives new conditions. In addition, it suggests a new combinatorial structure on tropicalizations of algebraic curves.
Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a certain condition involving a graph determined by these tropical curves, we can algorithmically find algebraic curves such that the tropicalization of their intersection is $D$.
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and intuition behind this limit and consider a few simple examples of correspondence principle between classical and tropical geometries.
This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring -- a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.
We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine notion of rank with the notion of tropical independence and has the property that the restriction of a tropical linear series of rank r to a connected subgraph is a tropical linear series of rank r. We show that tropical linear series of rank 1 are finitely generated as tropical modules and state a number of open problems related to algebraic, combinatorial, and …
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
Spherical tropical curves are balanced
本文证明了球面齐性空间 $G/H$ 中曲线的球面热带化满足平衡条件，推广了代数环面中曲线热带化的经典平衡性，并阐明了其与 Andreas Gross 平衡条件的关系。
Generic root counts of tropically transverse systems -- An invitation to tropical geometry in OSCAR
本文研究热带横截参数化多项式系统的通用解数，将其表示为修正系统的混合体积，并用初等热带几何学重新证明了耦合非线性振子平衡点数量的结果，同时介绍了OSCAR中的热带几何功能。
Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where…
Let E be a plane in an algebraic torus over an algebraically closed field. Given a balanced 1-dimensional fan C in the tropicalization of E, i.e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic…
We introduce the tropicalization of closed subschemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the results of Einslieder, Kapranov, Lind, Speyer and Sturmfels to higher local fields.
Algebraic loops, braids, and tropicalization
本文为光滑复代数簇上的代数环建立热带分类框架，证明其由热带点与初始簇连通分支决定，并应用于超平面排列补集，给出显式代表元及根赋值数据的热带解释。
Tropical vertex and real enumerative geometry
本文证明了精化热带顶点群中的交换子关系可由环面曲面上特定实有理曲线的枚举来刻画，建立了热带几何与实枚举几何之间的联系。
We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is shown that, for two generic preimages of the curves to an algebraic framework, their intersection projects exactly onto the stable intersection of the curves. It is also given sufficient conditions for such a generality in terms of the residual coefficients of the algebraic coefficients of defining equations of the curves.
Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as t…
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.
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…
Much like in the theory of algebraic geometry, we develop a correspondence between certain types of algebraic and geometric objects. The basic algebraic environment we work in is the a semifield of fractions H(x1,...,xn) of the polynomial semidomain H[x1,...,xn], where H is taken to be an idempotent semifield, while for the geometric environment we have the space H^n. We show that taking H to be idempotent makes both H(x1,...,xn) and Hn idempotent which turn out to satisfy many desired properties that we utilize for our construction. The fundamental algebraic and geometric objects having inter…
Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The aim of this paper is to generalise Kapranov's theorem, replacing the role of a valuation map, from a field to the real numbers union negative infinity, with a more general class of hyperfield homomorphisms, whose target is the tropical hyperfield and satisfy a relative algebraic closure condition. To provide an example of such a hyperfield homomorphism, the ma…
The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in several variables over complex numbers. The exposition is aimed for a general audience in mathematics and we hope to be accessible to undergraduate as well as advance high school students. The topics discussed belong to relatively new, and closely related branches of algebraic geometry which are usually referred to as tropical geometry and toric geometry. These ar…
In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for algebraic curves motivated by Berkovich's construction of skeletons of analytic curves. Under certain assumptions, we construct a one-to-one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects, called "stacky tropical reductions", that can be enumerated in terms of tropical curves satisfying linear c…
Given a tropical fan curve $Σ$ and a family of algebraic curves $X \rightarrow \mathbb{A}^k$ we define the realization locus $\mathop{Real}_Σ\subseteq \mathbb{A}^k$ as the set of fibers $X_a$ whose tropicalization is $Σ$. We produce an algorithm that describes the Zariski closure of $\mathop{Real}_Σ$ by imposing algebraic conditions for each ray of $Σ$.
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and covers of tropical curves maps between metric graphs satisfying certain conditions. In this short survey, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we showcase three results demonstrating how this theory can be applied in algebraic geometry.
We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind we prove a general correspondence theorem for enumerative problems which are defined via "evaluation maps" in both the algebraic and tropical world. Applying this to our motivational example we reprove Nishinou and Siebe…
This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4-8, 2011, and at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos Aires, August 1-5, 2011.
Tropicalization is a procedure that assigns polyhedral complexes to algebraic subvarieties of a torus. If one fixes a weighted polyhedral complex, one may study the set of all subvarieties of a toric variety that have that complex as their tropicalization. This gives a "tropical realization" moduli functor. We use rigid analytic geometry and the combinatorics of Chow complexes as studied by Alex Fink to prove that when the ambient toric variety is quasiprojective, the moduli functor is represented by a rigid space. As an application, we show that if a polyhedral complex is the tropicalization …
Quadratically Enriched Tropical Intersections
本文将热带几何方法应用于$\mathbb{A}^1$-枚举几何的丰富设置中，通过证明Bézout定理及其推广Bernstein-Kushnirenko定理，为在Grothendieck-Witt环中丰富的热带超曲面提供了理论基础。
We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective not-necessarily-normal toric varieties. The second part is a foundational account of tropical intersection theory with proofs of some new theorems relating it to classical intersection theory. Revised version includes many corrections, more examples, and improved exposition.
Let $G$ be a connected reductive algebraic group. We develop a Gröbner theory for multiplicity-free $G$-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space $G/H$. We define the notion of a spherical tropical variety and prove a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in $G/H$. Our work partly builds on the previous work of Vogiannou on spherical tropicalization and in some ways is complementary.
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 tropical geometry in the global setting using Berkovich's deformation retraction. We state and prove the generalized balancing conditions in this setting. Starting with a strictly semi-stable formal scheme, we calculate certain sheaves of vanishing cycles using analytic étale cohomology, then we interpret the tropical weights via these cycles. We obtain the balancing condition for tropical curves on the skeleton associated to the formal scheme in terms of the intersection theory on the special fiber. Our approach works over any complete discrete valuation field.
In this paper, we give some categorical description of the general spectrum functor, defining it as an adjoint of a global section functor. The general spectrum functor includes that of $F_1$ and of semirings.
The number of positive solutions of a system of two polynomials in two variables defined in the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. In this paper, we prove that a system as above constructed using this method has at most 6 positive solutions. We also show that this bound is sharp…
The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical polynomials, formed from univariate monomials, define subsemigroups with respect to coordinate-wise…
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.
In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid. We also prove a tropical local-to-global principle: Any closed, connected, locally tr…
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…
Space of prime congruences in tropical geometry
受经典概型理论启发，研究热带代数上的素同余空间及其几何，引入热带环面概型并给出素同余有限生成的条件。
Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d-1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.
The classical clutching and gluing maps between the moduli stacks of stable marked algebraic curves are not logarithmic, i.e. they do not induce morphisms over the category of logarithmic schemes, since they factor through the boundary. Using insight from tropical geometry, we enrich the category of logarithmic schemes to include so-called sub-logarithmic morphisms and show that the clutching and gluing maps are naturally sub-logarithmic. Building on the recent framework developed by Cavalieri, Chan, Wise, and the third author, we further develop a stack-theoretic counterpart of these maps in …
Lefschetz (1,1)-theorem in tropical geometry
本文证明热带流形上由热带循环基本类给出的 (n-1,n-1) 度热带同调类恰为特征波映射的核，通过建立有理多面体空间的热带 Lefschetz (1,1)-定理并结合庞加莱对偶性得出。
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) \cap Trop(X') lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(Δ) and its associated extended tropicalization N_R(Δ); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the …
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely d…
Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d,g) of complex plane curves of degree d and genus g through 3d+g-1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d,g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso-Harris formula in terms of both tropical geometry and lattice paths.
Tropical analogs of Milnor $K$-groups and tropicalizations of Zariski-Riemann spaces
本文引入有理 Milnor $K$-群的热带类似物并证明其 Zariski 层化具有 Gersten 分解，为后续建立平凡赋值域上光滑代数簇的热带 Hodge 猜想奠定基础。
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