Tropical Curves Research Articles

Pluricanonical cycles and tropical covers

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

Arithmetic counts of tropical plane curves and their properties

Abstract Recently, the first and third author proved a correspondence theorem which recovers the Levine- Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count ofplane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso- Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.

The volume polynomial of lattice polygons

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

Tropical curves in abelian surfaces I: enumeration of curves passing through points

Abstract This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.

Universality for tropical and logarithmic maps

We prove that every toric monoid appears in a space of maps from tropical curves to an orthant. It follows that spaces of logarithmic maps to Artin fans exhibit arbitrary toric singularities: a virtual universality theorem for logarithmic maps to pairs. The target rank depends on the chosen singularity: we show that the cone over the 7-gon never appears in a space of maps to a rank 1 target. We obtain similar results for tropical maps to affine space.

Tropical Refined Curve Counting with Descendants

We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _g$$\\end{document} class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.

Tropical Mirror

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the tropical Gromov-Witten invariants. We show that the sum over the amplitudes in $A$-model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the $A$-model HTQM by the mirror superpotential. We derived the mirror superpotentials for the toric varieties and showed that they coincide with the superpotentials in the mirror Landau-Ginzburg theory. We construct the mirror dual states to the evaluation observables in the tropical Gromov-Witten theory.

Linkage of graphs with flows

We prove several linkage properties of graphs with flows, generalizing some results on linkage of graphs. This translates in properties of connectedness through codimension one of certain posets. For example, the poset of flows and the posets of odd and even tropical spin curves. These posets are, respectively, the posets underlying the moduli space of roots of divisors on tropical curves and the moduli spaces of odd and even tropical spin curves.

Semiring isomorphisms between rational function semifields of tropical curves

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 T-algebra automorphism group of its rational function semifield, where T:=(R∪{−∞},max,+) is the tropical semifield. Finally, we describe all semiring automorphisms of rational function semifields of all tropical curves.

Tropical curves and solitons in nonlinear integrable systems

We construct families of rationally degenerating Riemann surfaces corresponding to tropical curves with trivial weights, and give explicit formulas of the associated quasi-periodic solutions and soliton solutions as their regularized limits to the KP hierarchy and the 2D Toda lattice hierarchy. This result especially gives rise to soliton solutions in extensive classes to these nonlinear integrable systems which are expected to cover all solutions expressed by compositions of elementary functions. For these quasi-periodic solutions, we also discuss their regularity and limits to the mixtures with solitons obtained from tropical curves with nontrivial weights.

Genus 0 logarithmic and tropical fixed‐domain counts for Hirzebruch surfaces

AbstractFor a non‐singular projective toric variety , the virtual logarithmic Tevelev degrees are defined as the virtual degree of the morphism from the moduli stack of logarithmic stable maps to the product . In this paper, after proving that Mikhalkin's correspondence theorem holds in genus 0 for logarithmic virtual Tevelev degrees, we use tropical methods to provide closed formulas for the case in which is a Hirzebruch surface. In order to do so, we explicitly list all the tropical curves contributing to the count.

The Ceresa class and tropical curves of hyperelliptic type

Abstract We define a new algebraic invariant of a graph G called the Ceresa–Zharkov class and show that it is trivial if and only if G is of hyperelliptic type, equivalently, G does not have as a minor the complete graph on four vertices or the loop of three loops. After choosing edge lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph G that is closely related to the Ceresa cycle for an algebraic curve defined over $\mathbb {C}(\!(t)\!)$ .

Tropical Moduli Spaces of Rational Graphically Stable Curves

The tropical moduli space $\mathcal{M}_{0,n}^{\textrm{trop}}$ is a cone complex which parameterizes leaf-labeled metric trees called tropical curves. We introduce graphic stability and describe a refinement of the cone complex given by radial alignment. We prove that given a complete multipartite graph $\Gamma$, the moduli space of radially aligned $\Gamma$-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 $\Gamma$.

Galois correspondence for Galois coverings of tropical curves

We define Galois coverings of tropical curves for which a Galois correspondence and a universal mapping property hold.

Topology of tropical moduli spaces of weighted stable curves in higher genus

Abstract We study the topology of moduli spaces of weighted stable tropical curves Δg ,w with fixed genus and unit volume. The space Δg ,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space M g ,w of weighted stable curves. When the genus is positive, we show that Δg ,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg ,w in terms of the combinatorics of the weight vector.

On moduli spaces of roots in algebraic and tropical geometry

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.

Periods of Tropical Curves and Associated KP Solutions

Periods of Tropical Curves and Associated KP Solutions

Tropical curves in abelian surfaces II: Enumeration of curves in linear systems

In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus g g curves in a fixed linear system of an abelian surface passing through g − 2 g-2 points to a tropical count. To do this, we relate the linear system defined by a complex curve to certain integrals of 1 1 -forms over cycles on the curve. We then give an expression for the tropical multiplicity provided by the correspondence theorem, and prove the invariance for the associated refined multiplicity, thus introducing refined invariants of Block-Göttsche type in abelian surfaces.

Plücker-Type Inequalities for Mixed Areas and Intersection Numbers of Curve Arrangements

Abstract Any collection of $n$ compact convex planar sets $K_1,\dots , K_n$ defines a vector of ${n\choose 2}$ mixed areas ${\operatorname {V}}(K_i,K_j)$ for $1\leq i<j\leq n$. We show that for $n\geq 4$ these numbers satisfy certain Plücker-type inequalities. Moreover, we prove that for $n=4$, these inequalities completely describe the space of all mixed area vectors $({\operatorname {V}}(K_i,K_j)\,:\,1\leq i<j\leq 4)$. For arbitrary $n\geq 4$, we show that this space has a semialgebraic closure of full dimension. As an application, we show that the pairwise intersection numbers of any collection of $n$ tropical curves satisfy the Plücker-type inequalities. Moreover, in the case of four tropical curves, any homogeneous polynomial relation between their six intersection numbers follows from the corresponding Plücker-type inequalities.

The tropical Poincaré-Hopf theorem

We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a “local” Poincaré-Hopf theorem). This provides another example of uncovering the “geometry” of matroids by expressing their invariants in terms of tropicalised geometric constructions. We also prove a global Poincaré-Hopf theorem and initiate the study of a more general tropical Lefschetz-Hopf trace formula by proving the two special cases of tropical curves and tropical tori.