The discriminant of a polynomial map is central to problems from affine geometry and singularity theory. Standard methods for characterizing it rely on elimination techniques that can often be ineffective. This paper concerns polynomial maps on the two-dimensional torus defined over a field of Puiseux series. We present a combinatorial procedure for computing the tropical curve of the discriminant of maps determined by generic polynomials with given supports. Our results enable one to compute the Newton polytope of the discriminant of complex polynomial maps on the plane.
Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any A-discriminant, without any smoothness assumption.
We study singularities in tropical hypersurfaces defined by a valuation over a field of positive characteristic. We provide a method to compute the set of singular points of a tropical hypersurface in positive characteristic and the p-adic case. This computation is applied to determine all maximal cones of the tropical linear space of univariate polynomials of degree $n$ and characteristic $p$ with a fixed double root and the fan of all tropical polynomials that have $0$ as a double root independently of the characteristic. We also compute, by pure tropical means, the number of vertices, edges…
We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface. i.e. we compute the real tropical discriminant.
We study A-discriminants from a non-Archimedean point of view, refining earlier work on the tropical discriminant. In particular, we study the case where $A$ is a collection of n+m+1 points in Z^n in general position, and give an algorithm to compute the image of the A-discriminant variety under the non-Archimedean evaluation map. When m=2, our approach yields tight lower and upper bounds, of order quadratic in n. We also detail a Sage package for plotting certain p-adic discriminant amoebae, and present explicit examples of point sets yielding discriminant amoebae with extremal behavior.
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve. Our approach is based on the close connection between analytic curves (in the sense of Berkovich) and tropical curves. We investigate the effect of these tropical modifications on the tropicalization map defined on the analytification of the given curve. Our study is motivated by the case of…
We study normal directions to facets of the Newton polytope of the discriminant of the Laurent polynomial system via the tropical approach. We use the combinatorial construction proposed by Dickenstein, Feichtner and Sturmfels for the tropicalization of algebraic varieties admitting a parametrization by a linear map followed by a monomial map.
In this paper we compute the Newton polytope $\mathcal M_A$ of the Morse discriminant in the space of univariate polynomials with the given support set $A.$ Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of $\mathcal M_A.$
Analogously as in classical algebraic geometry, linear pencils of tropical plane curves are parameterized by tropical lines in a coefficient space. A special example of such a linear pencil is the set of tropical plane curves with an n-element support set through a general configuration of n points in the tropical plane. In [RGST], it is proved that these linear pencils are compatible with their support set. In this article, we give a characterization of points lying in the fixed locus of a tropical linear pencil and show that each compatible linear pencil comes from a general configuration.
Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice points collinear. We prove that hyperelliptic graphs can only arise from such polygons. Along the way we will prove certain graphs do not embed tropically in the plane due to entirely combinatorial obstructions, regardless of whether their metric is actually hyperelliptic.
We determine necessary and sufficient conditions for real algebraic curves near the non-singular tropical limit to be hyperbolic with respect to a point, thus generalising Speyer's classification of stable curves near the tropical limit. In order to obtain the conditions, we develop tools of real tropical intersection theory. We introduce the tropical hyperbolicity locus and the signed tropical hyperbolicity locus of a real algebraic curve near the non-singular tropical limit. In the case of honeycombs, we characterise the tropical hyperbolicity locus in terms of the set of twisted edges on th…
We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these objects inside the tropical Grassmannian of planes in projective 5-space, and we explore a small tropical Hilbert scheme.
In this paper, we classify singular real plane tropical curves by means of subdivisions of Newton polytopes. First, we introduce signed Bergman fans (generalizing positive Bergman fans from [AKW06]) that describe real tropicalizations of real linear spaces ([Tab15]). Then, we establish a duality of real plane tropical curves and signed regular subdivisions of the Newton polytope and explore the combinatorics. We define a signed secondary fan that parametrizes real tropical Laurent polynomials and study the subset providing singular real plane tropical curves. A cone of the signed secondary fan…
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.
A smooth tropical quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig for lifting them to real bitangent lines are independent of the deformations. From this we deduce a tropical proof of Plücker and Zeuthen's count of the number of real bitangents to smooth plane quartic curves.
The tropical Grassmannian parameterizes tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\TP^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \leq 7$. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are used to draw pictures.
We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a tropical version of the above-mentioned result. In this work, we introduce stable tropical lines, which help in answering questions pertaining to incidence geometry in the tropical plane. Projective duality in the tropical plane helps in translating the question for stable lines to stable intersections that have been previously studied in depth. Invoking dua…
Several results in tropical geometry have related the j-invariant of an algebraic plane curve of genus one to the cycle length of a tropical curve of genus one. In this paper, we prove that for a plane cubic over the field of Puiseux series the negative of the generic valuation of the $j$-invariant is equal to the cycle length of the tropicalization of the curve, if there is a cycle at all.
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.
We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with $g$ interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$. We compute these spaces explicitly for $g \leq 5$.
Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus $12$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most $11$. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all …
Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus $3$ can be realized as a tropicalization of a quartic in the euclidean plane. In this article, we focus on the interior of the maximal cones in the moduli space and classify all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, we show that these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropica…
We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Final…
We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.
In analogy with the classical group law on a plane cubic curve, we define a group law on a smooth plane tropical cubic curve. We show that the resulting group is isomorphic to $S^1$.
We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in $\mathbb{TP}^5$. This follows from a result of Sturmfels and Speyer which classifies these tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian $\operatorname{Gr}(3,6)$. Speyer and Williams show that the positive part $\operatorname{Gr}^+(3,6)$ of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type $D_4$. We provide a st…
Finding the so-called characteristic numbers of the complex projective plane ${\mathbb C}P^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is $3d-1+g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding en…
In this paper we fully describe all tropical linear mappings in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by an order 3 matrix. An erratum has been added fixing two mistakes in the main paper.
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.
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 $D$.
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…
In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemannian surfaces.
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.
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.
In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point con…
The paper consists of lecture notes for a mini-course given by the authors at the Gökova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.
We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. As a consequence, we prove that the tropical multiplicities of all points in the tropical projective Grassmannian are equal to one. Finally, we determine a piec…
For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity and show the existence of a tropical curve satisfying a certain formula, which looks like a well-known formula for a real morsification due to A'Campo and Gusein-Zade.
In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R.Pandharipande using Gromov-Witten theory. In this article we treat the tropical analogue of this problem, the determination of the number of tropical elliptic plane curves of degree d and fixed ``tropical j-invariant'' interpolating an appropriate number of points in general position. We show that this number is independent of the position of the points and the value of the j-invariant and that it coincides with the number of complex elliptic curves.…
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.
We propose an iterated version of the Gilbert model, which results in a sequence of random mosaics of the plane. We prove that under appropriate scaling, this sequence of mosaics converges to that obtained by a classical Poisson line process with explicit cylindrical measure. Our model arises from considerations on tropical plane curves, which are zeros of random tropical polynomials in two variables. In particular, the iterated Gilbert model convergence allows one to derive a scaling limit for Poisson tropical plane curves. Our work raises a number of open questions at the intersection of sto…
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…
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…
What polynomial in the coefficients of a system of algebraic equations should be called its discriminant? We prove a package of facts that provide a possible answer. Let us call a system typical, if the homeomorphic type of its set of solutions does not change as we perturb its (non-zero) coefficients. The set of all atypical systems turns out to be a hypersurface in the space of all systems of k equations in n variables, whose monomials are contained in k given finite sets. This hypersurface B contains all systems that have a singular solution, this stratum is conventionally called the discri…
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.
Every graph $Γ$ can be embedded in the plane with a minimal number of edge intersections, called its classical crossing number $\text{cross}\left(Γ\right)$. In this paper, we prove that if $Γ$ is a metric graph it can be realized as a tropical curve in the plane with exactly $\text{cross}\left(Γ\right)$ crossings, where the tropical curve is equipped with the lattice length metric. Our result has an application in algebraic geometry, as it enables us to construct a rational map of non-Archimedean curves into the projective plane, whose tropicalization is almost faithful when restricted to thei…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to significant progress. In this survey, we give an introduction to tropical geometry techniques for algebraic curve counting problems. We also survey some recent developments, with a particular emphasis on the computation of the degree of the Severi varieties of the complex projective plane and other toric surfaces as well as Hurwitz numbers and applications to rea…
We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but the only known proof was indirect with the help of Mikhalkin's Correspondence Theorem that translates this question into the well-known fact that the numbers of complex curves in a toric surface through some given points do not depend on the position of the points. This paper presents a direct proof en…
