A tropical curve Γis a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γanalogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from Γto a tropical projective space, and the image can be extended to a tropical curve of degree equal to °(D). The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a \Q-tropical curve is a direct limit of critical groups of finite graphs converging to the curve.
We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends only on the underlying graph of a tropical curve and is preserved when passing to genus $\geq 2$ connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.
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 prove invariance for the number of planar tropical curves enhanced with polynomial multiplicities recently proposed by Florian Block and Lothar Goettsche. This invariance has a number of implications in tropical enumerative geometry.
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…
In this article we introduce the recently developed polymake extension TropicalQuarticCurves and its associated database entry in polyDB dealing with smooth tropical quartic curves. We report on algorithms implemented to analyze tropical bitangents and their lifting conditions over real closed valued fields. The new functions and data were used by the authors to provide a tropical proof of Plücker and Zeuthen's count of real bitangents to smooth quartic curves.
We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a non-singular fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no …
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems.
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 $Γ$.
A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected.
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…
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.
We introduce the notion of tropical area of a tropical curve defined in an open subset of $\mathbb R^n$. We prove that the number of vertices of a tropical curve is bounded by the area of the curve. The approach is totally elementary yet tricky. Our proof employs ideas from intersection theory in algebraic geometry. The result can be interpreted as the fact that the moduli space of tropical curves with bounded area is of finite type.
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.
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…
We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus $g$. This space parametrizes equivalence classes of tropical curves of genus $g$ together with a $μ$-polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decomposit…
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 $S^1$.
By tropical Abel-Jacobi theorem, the Jacobian of a tropical curve is isomorphic to the Picard group. A tropical curve in $\mathbb{R}^2$ corresponds to an immersion from a tropical curve to $\mathbb{R}^2$. In this paper, we show that any principal divisor on a tropical curve is the restriction of a principal divisor on the ambient plane $\mathbb{R}^2$.
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.
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the resulting weighted metric graph. In general, it is not known how to find the abstract tropicalization of a curve defined by polynomial equations, since an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction in practice. We solve this problem in the case of hyperelliptic curves by studying admissible co…
On tropical geometry in $\rr^2$, the divisor and the Jacobian variety are defined in analogy to algebraic geometry. For study of these objects, it is important to think of the `bunch' of a tropical curve. In this paper, we will show that if the bunch is a bouquet, then the Jacobian is a higher-dimensional torus.
We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill-Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill-Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill-Noether rank of a tropical curve with the dimension of the Brill-Noether lo…
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using the development of ``mechanism'' which is based on ``distortion'' values and their matrices, we discuss some aspects refereing to quadrics with respect to their dual objects. This topic includes also the induced dual subdivision of Newton Polytope and its compatible properties. Finally, a regularity of tropical curves in the duality sense is generally defined …
We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on the associated minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of P^1 with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a fait…
In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear tropical varieties from Ardila and Klivans and from Feichtner and Sturmfels. We show that a singularity tropicalises either to a vertex of higher valence or of higher multiplicity, or to an edge of higher weight. We then classify maximal dimensional types of singular tropical curves. For those, the singularity is either a crossing of two edges, or a 3-valent ver…
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.
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 define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices; we prove that the Riemann-Roch theorem holds in both cases. We extend Baker's Specialization Lemma to weighted graphs.
We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g-1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g-1 vertices with maximum valence 3. Finally, we s…
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
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.
We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves, similar to the recent proof of the Riemann-Roch theorem for graphs by Baker and Norine, is presented. In addition, a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph is confirmed, and an algorithm for computing the rank of a divisor on a tropical curve is constructed.
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 …
Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Fina…
Given a divisor $D$ on a tropical curve $Γ$, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system $|D|$. This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the linear system, and in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points on tropi…
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and tropical Chern cycles and numbers. We provide a new method for constructing tropical surfaces, called the tropical sum, similar to the fiber sum of usual manifolds. We prove a tropical adjunction formula for curves in compact tropical surfaces satisfying a local condition, a partial Castelnuovo-Enriques criterion for contracting (-1)-curves, and also invariance of …
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…
For a projective curve $C$ and the canonical divisor $K_C$ on $C$, it is classically known that the canonical ring $R(C) = \oplus_{m=0}^\infty H^0(C, m K_C)$ is finitely generated in degree at most three. In this article, we study whether analogous statements hold for finite graphs and tropical curves. For any finite graph $G$, we show that the canonical semi-ring $R(G)$ is finitely generated but that the degree of generators are not bounded by a universal constant. For any hyperelliptic tropical curve $Γ$ with integer edge-length, we show that the canonical semi-ring $R(Γ)$ is not finitely ge…
A $K_4$-curve is a smooth, proper curve X of genus 3 over a nonarchimedean field whose Berkovich skeleton $Γ$ is a complete graph on 4 vertices. The curve X has 28 effective theta characteristics, i.e. the 28 bitangents to a canonical embedding, while $Γ$ has exactly seven tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a $K_4$-curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.
We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a tropical intersection-theoretic definition of tropical triple Hurwitz numbers. We show that our intersection-theoretic definition coincides with the one given by Bertrand, Brugallé and Mikhalkin in the article "Tropical Open Hurwitz numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus …
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 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…
We introduce a tropical geometric framework that allows us to define $ψ$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $ψ$ classes for some one-dimensional families of genus-one tropical curves.
We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry fo…
We investigate geometric embeddings among several classes of stacky fans and algorithms, e.g., to compute their homology. Interesting cases arise from moduli spaces of tropical curves. Specifically, we show that the tropical honeycomb curves form a contractible sub-locus in the moduli of all tropical $K_4$-curves.
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…
The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with delta nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable y, which are conjecturally equal, for large d. At y = 1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our desc…
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…
In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of n-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for n \leq 6. In the second part of the paper, we show that the locus of point configurations in (\R^2)^n in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension 1. In addition, we compute explicitly the weights of this cycle.
The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is bas…
We define tropical Psi-classes on the moduli space of rational tropical curves in R^2 and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions.
