We contribute to the foundations of tropical geometry with a view towards 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.
The jaggedness of an order ideal I in a poset P is the number of maximal elements in I plus the number of minimal elements of P not in I . A probability distribution on the set of order ideals of P is toggle-symmetric if for every p ∈ P, the probability that p is maximal in I equals the probability that p is minimal not in I . In this paper, we prove a formula for the expected jaggedness of an order ideal of P under any toggle-symmetric probability distribution when P is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan-López-Pflueger-Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp-Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.
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 lies in a maximal cell of genus g called a standard ladder. Definitions and notationWe start by defining harmonic morphisms, quotients, divisors, and rational functions on metric graphs. These concepts will be used in Theorem 3.13 to characterize hyperellipticity. §2.1. Metric graphs and harmonic morphisms. Throughout, all of our graphs are connected, with loops and multiple edges allowed. A metric graph is a metric space Γ such that
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.