Moduli Space Of Tropical Curves Research Articles

Tropical ψ classes

We introduce a tropical geometric framework that allows us to define $\psi$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $\psi$ classes for some one-dimensional families of genus-one tropical curves.

Schottky spaces and universal Mumford curves over $${\mathbb {Z}}$$

For every integer \(g \ge 1\) we define a universal Mumford curve of genus g in the framework of Berkovich spaces over \({\mathbb {Z}}\). This is achieved in two steps: first, we build an analytic space \({{\mathcal {S}}}_g\) that parametrizes marked Schottky groups over all valued fields. We show that \({{\mathcal {S}}}_g\) is an open, connected analytic space over \({\mathbb {Z}}\). Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of \({{\mathcal {S}}}_g\), both locally and globally. As a result, we can define the universal Mumford curve \({{\mathcal {C}}}_g\) as a relative curve over \({{\mathcal {S}}}_g\) such that every Schottky uniformized curve can be described as a fiber of a point in \({{\mathcal {S}}}_g\). We prove that the curve \({{\mathcal {C}}}_g\) is itself uniformized by a universal Schottky group acting on the relative projective line \(\mathbb {P}^1_{{{\mathcal {S}}}_g}\). Finally, we study the action of the group \(\hbox {Out}(F_g)\) of outer automorphisms of the free group with g generators on \({{\mathcal {S}}}_g\), describing the quotient \(\hbox {Out}(F_g) \backslash {{\mathcal {S}}}_g\) in the archimedean and non-archimedean cases. We apply this result to compare the non-archimedean Schottky space with constructions arising from geometric group theory and the theory of moduli spaces of tropical curves.

Tropicalization of the universal Jacobian

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.

The moduli space of tropical curves with fixed Newton polygon

Abstract Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. If this polygon is nonhyperelliptic or maximal and hyperelliptic, then this formula matches the dimension of the moduli space of nondegenerate algebraic curves with the given Newton polygon.

Moduli dimensions of lattice polygons

Given a lattice polygon P with g interior lattice points, we can associate to \(P\) two moduli spaces: the moduli space of algebraic curves that are non-degenerate with respect to \(P\) and the moduli space of tropical curves of genus g with Newton polygon P. We completely classify the possible dimensions such a moduli space can have in the tropical case. For non-hyperelliptic polygons, the dimension must be between g and \(2g+1\) and can take on any integer value in this range, with exceptions only in the cases of genus 3, 4, and 7. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from g to \(2g-1\). In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to P.

Tropical superelliptic curves

Abstract We present an algorithm for computing the Berkovich skeleton of a superelliptic curve yn = f(x) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when n is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus g.

A MODULI STACK OF TROPICAL CURVES

We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

Non-Archimedean geometry of Artin fans

The purpose of this article is to study the role of Artin fans in tropical and non-Archimedean geometry. Artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato's notion of a fan. Every logarithmic algebraic stack admits a tautological strict morphism ϕX:X→AX to an associated Artin fan. The main result of this article is that, on the level of underlying topological spaces, the natural functorial tropicalization map of X is nothing but the non-Archimedean analytic map associated to ϕX by applying Thuillier's generic fiber functor. Using this framework, we give a reinterpretation of the main result of Abramovich–Caporaso–Payne identifying the moduli space of tropical curves with the non-Archimedean skeleton of the corresponding algebraic moduli space.

Integral Homology of the Moduli Space of Tropical Curves of Genus 1 with Marked Points

Integral Homology of the Moduli Space of Tropical Curves of Genus 1 with Marked Points

Tropicalization of the moduli space of stable maps

Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in $X$, the associated tropical curves in $S$ deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination for rigid subanalytic sets.

The number of vertices of a tropical curve is bounded by its area

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.

Combinatorics of the tropical Torelli map

This paper is a combinatorial and computational study of the moduli space of tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were introduced recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which the aforementioned moduli spaces are objects and the Torelli map is a morphism. We compute the poset of cells of tropical M_g and of the tropical Schottky locus for genus at most 5. We show that tropical A_g is Hausdorff, and we also construct a finite-index cover for A_3 which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive semidefinite forms, and metric graphs, play a role.

Tropical hyperelliptic curves

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 show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.

Geometry of tropical moduli spaces and linkage of graphs

We prove the following “linkage” theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.

Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits

The main characters of this paper are the moduli spaces TMg,n of tropical curves of genus g with n marked points, with g ≥ 2. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces Xg,n, which in turn possess natural representations as homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes Δg, with the latter being interesting on their own right.

On the tropical Torelli map

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.

Tropical fans and the moduli spaces of tropical curves

AbstractWe give a rigorous definition of tropical fans (the ‘local building blocks for tropical varieties’) and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some ℝr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any ℝr) through given points are independent of the points.

The Topology of Moduli Spaces of Tropical Curves with Marked Points

The Topology of Moduli Spaces of Tropical Curves with Marked Points

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces M G 1 , n v , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces T M 1 , n , which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.