The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the ``lifting algorithm'' using Singular and Gfan if the base field are the rational numbers. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K^{n+1},0).
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 $D$.
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 …
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 complete real-valued field $k$ of residue characteristic zero, we study properties of a meromorphic $q$-differential form $ξ$ (a section of $Ω_X^{\otimes q}$) on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,ξ)$ a natural tropical reduction datum combining tropical-geometric data over the value group of $|k^\times|$ and algebro-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility conditions, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actua…
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational weighted graphs in tropical surfaces. Specifically, we study the `lifting' problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a surface? We develop specific combinatorial obstructions to lifting a graph by reducing the problem to the question of whether or not one can factor a polynomial with particular support in the cha…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral complex and compute its dimension. When the genus is at most 1, we show that all the tropical divisors that move in the expected dimension are realizable. As part of the proof, we introduce a combinatorial tool for explicitly constructing large families of realizable tropical divisors.
Let $k$ be a field with a real valuation $ν$ and $R$ a $k$-algebra. We show that there exist a $k$-algebra $K$ and a real valuation $μ$ on $K$ extending $ν$ such that any real ring valuation of $R$ is induced by $μ$ via some homomorphism from $R$ to $K$; $K$ can be chosen to be a field. Then we study the case when $ν$ is trivial and $R$ a complete local Noetherian ring with the residue field $k$. Let $K$ be the ring $\bar{k}[[t^\R]]$ of Hahn series with its natural valuation $μ$; $\bar{k}$ is an algebraic closure of $k$. Despite $K$ is not universal in the strong sense defined above, it has th…
We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents, however the conditions to de…
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mnëv universality with explicit bounds on the size of the matroid, which may be of independent interest.
In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation g…
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1. The proofs require not only the geometry of compactified tropicalizations of subvarieties of toric varieties, but also new results about the geometry of finite type schemes over non-noetherian valuation rings of …
Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a differential form $ω$ on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,ω)$ a natural tropical reduction datum combining tropical data of $(X,ω)$ and algebra-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility condition, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair $(X,ω)$. In particular, we obtain a short proof of the main result of a work [B…
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 …
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…
We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity of the ideal is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on the depth of the ideal from the fan structure of the generic tropical variety if the depth is known to be greater than 0. In particular, in this case we can see if the ideal is Cohen-Macaulay or almost-C…
We show that in the constant coefficient case the generic tropical variety of a graded ideal exists. This can be seen as the analogon to the existence of the generic initial ideal in Groebner basis theory. We determine the generic tropical variety as a set in general and as a fan for principal ideals and linear ideals.
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.
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: Where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying constraints on the images of classical intersections, and, second, showing that all tropical configurations satisfying these constraints can be achieved. This paper provides the first part: images of intersection points must be linearly equivalent to the stable tropical intersection by a suitable rational function. Several examples provide evidence for the con…
Each Gröbner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gröbner stratification is a coarsening of the decomposition into Gröbner polyhedra, and has the advantage that it does not depend on a choice of compactification. We give an example of a curve over a field with non-trivial valuation whose Gröbner stratification is strictly finer than the coarsest polyhedral decomposition of the tropical variety. We also show that the Gröbner stratification of a locally matroidal tropical variety is completely determined by the underlying…
In earlier papers it was shown that the generic tropical variety of an ideal can contain information on algebraic invariants as for example the depth in a direct way. The existence of generic tropical varieties has so far been proved in the constant coefficient case for the usual notion of genericity. In this paper we generalize this existence result to include the case of non-constant coefficients in certain settings. Moreover, we extend the notion of genericity to arbitrary closed subvarieties of the general linear group. In addition to including the concept of genericity on algebraic groups…
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.
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…
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.
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…
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.
The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety in a tropicalised toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. The Chow polytope of a tropical variety $X$ is given by a simple combinatorial construction: its normal subdivision is the Minkowski sum of $X$ and a reflected skeleton of the fan of the ambient toric variety.
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…
We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety.…
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…
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.
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.
The hypertoric variety $\mathfrak{M}_{\mathcal{A}}$ defined by an affine arrangement $\mathcal{A}$ admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.
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 …
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…
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.
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…
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.
