Moduli Spaces of Rational Weighted Stable Curves and Tropical Geometry

Abstract

We study moduli spaces of rational weighted stable tropical curves, and their connections with 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, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

“…In , as an archetypical result, the authors show that the moduli space of tropical curves is isomorphic to the skeleton of the moduli space of algebraic curves with the respect to the toroidal structure coming from the Deligne–Knudsen–Mumford compactification (see and ). Similar results have appeared for other moduli spaces, such as the moduli space of admissible covers (see ), the moduli space of weighted stable curves (see for the case of genus and for ), and the moduli space of rational logarithmic stable maps into a toric variety (see ).…”

The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

“…Proof. The fact that J d,g,1 ⊂ J μ,g and the map J μ,g → M g,1 are toroidal follows from equations (15) and (16), and from the local description of the map π.…”

Tropical geometry of moduli spaces of weighted stable curves

“…During the work on this project, the author learned about the article [8], which contains the g = 0 case of Theorem 1.2 in [8,Theorem 3.15]. The main goal of [8], however, is to treat the tropicalization of M 0,A from the point of view of geometric tropicalization, as developed in [15] and further studied in [11]. For this, the authors of [8] embed M 0,A into a toric variety X, and study the tropicalization Trop X (M 0,A ) of M 0,A with respect to X.…”

“…The main goal of [8], however, is to treat the tropicalization of M 0,A from the point of view of geometric tropicalization, as developed in [15] and further studied in [11]. For this, the authors of [8] embed M 0,A into a toric variety X, and study the tropicalization Trop X (M 0,A ) of M 0,A with respect to X. By [35, Theorem 1.1 and 1.2] as well as Theorem 1.2, there is a natural continuous and surjective map M trop 0,A −→ Trop X (M 0,A ), that is, in general, not injective, as can be seen in [8,Figure 4].…”

“…For this, the authors of [8] embed M 0,A into a toric variety X, and study the tropicalization Trop X (M 0,A ) of M 0,A with respect to X. By [35, Theorem 1.1 and 1.2] as well as Theorem 1.2, there is a natural continuous and surjective map M trop 0,A −→ Trop X (M 0,A ), that is, in general, not injective, as can be seen in [8,Figure 4]. The main result of [8] is a characterization of those weights A for which this map is a bjiection, that is, for which the geometric tropicalization of M 0,A faithfully represents the full tropical moduli space M trop 0,A .…”