Tropical Curve Research Articles

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.

The distribution of Weierstrass points on a tropical curve

The distribution of Weierstrass points on a tropical curve

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 $\mathbb {C}(\!(t)\!)$ .

The entropy of the radical ideal of a tropical curve

The entropy of a semiring ideal of tropical polynomials is introduced. The radical of a semiring ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with zero coefficients vanishes. Also, we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical semiring ideal with a positive entropy is exhibited.

Abelian tropical covers

AbstractLet $\mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $\mathfrak{A}$ -covers of a tropical curve $\Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $\Gamma$ . We give a realisability criterion for harmonic $\mathfrak{A}$ -covers by patching local monodromy data in an extended homology group on $\Gamma$ . As an explicit example, we work out the case $\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$ and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.

Some statistics about Tropical Sandpile Model

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model represents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ of points in a compact convex domain $\Omega\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_\Omega$. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number $n$ of randomly dropped points $P=\{p_1,\dots,p_n\}\subset[0,1]^2=\Omega$ and the degree of the tropical polynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve $C(G_{P}{\bf 0}_\Omega)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$. The main theoretical result is that $C(G_{P}{\bf 0}_\Omega)\setminus (P\cap \partial\Omega)$, i.e. the tropical curve in $\Omega^\circ$ with marked points $P$ removed, is a tree.

Rational function semifields of tropical curves are finitely generated over the tropical semifield

We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield [Formula: see text] by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves [Formula: see text], the rational function semifield of [Formula: see text] is finitely generated as a [Formula: see text]-algebra, where [Formula: see text] stands for the pull-back of the rational function semifield of [Formula: see text] by [Formula: see text].

Intersecting psi-classes on tropical Hassett spaces

We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig for tropical moduli spaces of rational stable curves with distinct marked points. Our computation reveals that the weight of a maximal cone in an intersection has a combinatorial intepretation in terms of the underlying tropical curve and it is always nonnegative. In particular, our result specialises to that, in top dimension, the tropical intersection product coincides with its classical counterpart.

Galois quotients of tropical curves and invariant linear systems

For a map $\varphi : \Gamma \rightarrow \Gamma'$ between tropical curves and an isometric action on $\Gamma$ of a finite group $K$, $\varphi$ is a $K$-Galois covering on $\Gamma'$ if $\varphi$ is a harmonic morphism, the degree of $\varphi$ coincides with the order of $K$ and the action of $K$ induces a transitive action on every fibre. We prove that for a tropical curve $\Gamma$ with an isometric action of a finite group $K$, there exists a rational map, from $\Gamma$ to a tropical projective space, which induces a $K$-Galois covering on the image with proper edge-multiplicities. As an application, we also prove that for a hyperelliptic tropical curve without one valent points and of genus at least two, the invariant linear system of the hyperelliptic involution $\iota$ of the canonical linear system, the complete linear system associated with the canonical divisor, induces an $\langle \iota \rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.

Symmetric powers of algebraic and tropical curves: A non-Archimedean perspective

We show that the non-Archimedean skeleton of the d d -th symmetric power of a smooth projective algebraic curve X X is naturally isomorphic to the d d -th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X 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 certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.

Tropical floor plans and enumeration of complex and real multi-nodal surfaces

The family of complex projective surfaces in P 3 \mathbb {P}^3 of degree d d having precisely δ \delta nodes as their only singularities has codimension δ \delta in the linear system | O P 3 ( d ) | |{\mathcal O}_{\mathbb {P}^3}(d)| for sufficiently large d d and is of degree N δ , C P 3 ( d ) = ( 4 ( d − 1 ) 3 ) δ / δ ! + O ( d 3 δ − 3 ) N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)=(4(d-1)^3)^\delta /\delta !+O(d^{3\delta -3}) . In particular, N δ , C P 3 ( d ) N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d) is polynomial in d d . By means of tropical geometry, we explicitly describe ( 4 d 3 ) δ / δ ! + O ( d 3 δ − 1 ) (4d^3)^\delta /\delta !+O(d^{3\delta -1}) surfaces passing through a suitable generic configuration of n = ( d + 3 3 ) − δ − 1 n=\binom {d+3}{3}-\delta -1 points in P 3 \mathbb {P}^3 . These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface S S which are sufficient to reconstruct S S . In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration w \boldsymbol {w} of real points, the number N δ , R P 3 ( d , w ) N_{\delta ,\mathbb {R}}^{\mathbb {P}^3}(d,\boldsymbol {w}) of real surfaces of degree d d having δ \delta real nodes and passing through w \boldsymbol {w} is bounded from below by ( 3 2 d 3 ) δ / δ ! + O ( d 3 δ − 1 ) \left (\frac {3}{2}d^3\right )^\delta /\delta ! +O(d^{3\delta -1}) . We prove analogous statements for counts of multinodal surfaces in P 1 × P 2 \mathbb {P}^1\times \mathbb {P}^2 and P 1 × P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1 .

Scattering diagrams, theta functions, and refined tropical curve counts

Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of Fock and Goncharov. We also prove a refined version of the Carl-Pumperla-Siebert result on consistency of theta functions, and we prove the non-degeneracy of the trace-pairing for the Gross-Hacking-Keel Frobenius structure conjecture.

Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

Generators of invariant linear system on tropical curves for finite isometry group

For a tropical curve $\Gamma$ and a finite subgroup $K$ of the isometry group of $\Gamma$, we prove, extending the work by Haase, Musiker and Yu ([6]), that the $K$-invariant part of the complete linear system associated to a $K$-invariant effective divisor on $\Gamma$ is finitely generated

(Symplectic) leaves and (5d Higgs) branches in the Poly(go)nesian Tropical Rain Forest

We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.

Realizability of tropical canonical divisors

We use recent results by Bainbridge–Chen–Gendron–Grushevsky–Möller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair (\Gamma, D) consisting of a stable tropical curve \Gamma and a divisor D in the canonical linear system on \Gamma , we give a purely combinatorial condition to decide whether there is a smooth curve X over a non-Archimedean field whose stable reduction has \Gamma as its dual tropical curve together with an effective canonical divisor K_X that specializes to D .

Spines for amoebas of rational curves

To every rational complex curve C \subset (\mathbb C^\times)^n we associate a rational tropical curve \Gamma \subset \mathbb R^n so that the amoeba \mathcal A(C) \subset \mathbb R^n of C is within a bounded distance from \Gamma . In accordance with the terminology introduced in [PR], we call \Gamma the spine of \mathcal A(C) . We use spines to describe tropical limits of sequences of rational complex curves.

Tropical curves of hyperelliptic type

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 $$\ge 2$$ connected minors. The main result is an forbidden minors characterization of tropical curves of hyperelliptic type.

Tropically constructed Lagrangians in mirror quintic threefolds

AbstractWe use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds.We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve.As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that$>300$mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves. For higher-genus situations, we prove the correspondence for the non-superabundant part of the invariant. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.