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 tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve $C$ and a divisor $D$ of non-negative rank on $C$, reduced divisors equivalent to $D$ define a morphism from $C$ to the complete linear system $|D|$, which is described in terms of Wronskians.
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 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.
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 his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex $Γ$ is a $Δ$-complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors that are l…
We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact tropical convex sets. In this paper, we build the foundation of a theory of idempotent analysis using tropical projections and obtain a series of subsequent results, e.g. tropical retracts, construction of compact tropical convex sets and a set-theoretical characterization of tropical weak independence. In particular, we prove a tropical version of Mazur's Theor…
We consider a variant of metrised graphs where the edge lengths take values in a commutative monoid, as a higher-rank generalisation of the notion of a tropical curve. Divisorial gonality, which Baker and Norine defined on combinatorial graphs in terms of a chip firing game, is extended to these monoid-metrised graphs. We define geometric gonality of a monoid-metrised graph as the minimal degree of a horizontally conformal, non-degenerate morphism onto an monoid-metrised tree, and prove that geometric gonality is an upper bound for divisorial gonality in the monoid-metrised case. We also show …
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 $\mathbb{R}^2$.
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…
Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the "volume" of each divisor in the cone - that is, the degree of its top power - is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with n…
Tropical Weierstrass points and Weierstrass weights
CNRS - Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay（法国国家科学研究中心-奥赛数学实验室，巴黎萨克雷大学）；Institut de mathématiques de Marseille, Université d’Aix-Marseille, CNRS, Centrale Marseille（马赛数学研究所，艾克斯-马赛大学，法国国家科学研究中心，中央马赛）；Department of Mathematics, University of Washington — Mathematics Division, National Center for Theoretical Sciences（华盛顿大学数学系，国家理论科学中心）
Using a potential theory on metric graphs "Gamma", we introduce the notion of tropical convexity to the space "RDiv^d(Gamma)" of effective R-divisors of degree d on "Gamma" and show that a natural metric can be defined on "RDiv^d(Gamma)". In addition, we extend the notion of reduced divisors which is conventionally defined in a complete linear system |D| with respect to a single point in "Gamma". In our general setting, a reduced divisor is defined uniquely as an R-divisor in a compact tropical convex subset "T" of "RDiv^d(Gamma)" with respect to a certain R-divisor "E" of the same degree d. I…
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…
Let $Γ$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $Γ$. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an "integral" version of this result which is of independent interest. As an application, we provide a "geometric proof" of (a dual version of) Kirchhoff's celebrated Matri…
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…
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.
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.
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 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 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…
This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical $ψ$-class divisors.
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…
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…
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…
Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; this proof might be generalisable to cover all Veronese embeddings, whose secant dimensions are known from the ground-breaking but difficult work of Alexander and Hirschowitz. Also, the non-defectiveness of certain Segre embeddings is proved, w…
Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $Δ_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $Δ_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $Δ_{g, w}$ is simply connected for all values of $w$. We also give a formula for the Euler characteristic of $Δ_{g, w}$ in terms of the combinatorics of $w$.
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.
We compute the automorphism group $\mathrm{Aut}(Δ_{g, n})$ for all $g, n \geq 0$ such that $3g - 3 + n > 0$, where $Δ_{g, n} \subset M_{g, n}^\mathrm{trop}$ is the moduli space of stable $n$-marked tropical curves of genus $g$ and volume one. In particular, we show that $\mathrm{Aut}(Δ_{g})$ is trivial for $g \geq 2$, while $\mathrm{Aut}(Δ_{g, n}) \cong S_n$ when $n \geq 1$ and $(g, n) \neq (0, 4), (1, 2)$. The space $Δ_{g, n}$ is a symmetric $Δ$-complex in the sense of Chan, Galatius, and Payne, and is identified with the dual intersection complex of the boundary divisor in the Deligne-Mumfor…
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 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…
Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometr…
Let $X$ be a closed algebraic subset of $\mathbb{A}^{n}(K)$ where $K$ is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich space of $X$ to an inverse limit of a certain family of embeddings of $X$ called linear tropicalizations of $X$. This map is injective on the subset of the Berkovich space $X^{an}$ which contains all seminorms arising from closed points of $X$. We show that the map is a homeomorphism if $X$ is a non-singular algebraic curve. Some applications of these results t…
The moduli space $Δ_{g,w}$ of tropical $w$-weighted stable curves of volume $1$ is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of $w$-weighted stable curves. If at least two of the weights are $1$, we prove that $Δ_{0,w}$ is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space $Δ_{0,w}$ is disconnected and for which the fundamental group of $Δ…
We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry.
We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P\neq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.
The linear system $|D|$ of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute its extremal generators using the landmarks. We apply these methods to some examples - namely the canonical linear systems of some small trivalent graphs.
