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Tropical curves and metric graphs

  • Author(s): Chan, Melody Tung
  • Advisor(s): Sturmfels, Bernd
  • et al.

In just ten years, tropical geometry has established itself as an important new field bridging algebraic geometry and combinatorics whose techniques have been used to attack problems in both fields. It also has important connections to areas as diverse as geometric group theory, mirror symmetry, and phylogenetics. Our particular interest here is the tropical geometry associated to algebraic curves over a field with nonarchimedean valuation. This dissertation examines tropical curves from several angles.

An abstract tropical curve is a vertex-weighted metric graph satisfying certain conditions (see Definition 2.2.1), while an embedded tropical curve takes the form of a 1-dimensional balanced polyhedral complex in R^n. Both combinatorial objects inform the study of algebraic curves over nonarchimedean fields. The connection between the two perspectives is also very rich and is developed e.g. in [Pay09] and [BPR11]; we give a brief overview in Chapter 1 as well as a contribution in Chapter 4.

Chapters 2 and 3 are contributions to the study of abstract tropical curves. We begin in Chapter 2 by studying the moduli space of abstract tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map, as initiated in [BMV11]. We provide a detailed combinatorial and computational study of these objects and 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.

In Chapter 3, we study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. Our work ties together two strands in the tropical geometry literature, namely the study of the tropical moduli space of curves and tropical Brill-Noether theory. Our methods are graph-theoretic and extend much of the work of Baker and Norine [BN09] on harmonic morphisms of graphs to the case of metric graphs. We also provide new computations of tropical hyperelliptic loci in the form of theorems describing their specific combinatorial structure.

Chapter 4 presents joint work with Bernd Sturmfels and is a contribution to the study of tropical curves as balanced embedded 1-dimensional polyhedral complexes. We say that a plane cubic curve, defined over a field with valuation, is in honeycomb form if its tropicalization exhibits the standard hexagonal cycle shown in Figure 4.1. We explicitly compute such representations from a given j-invariant with negative valuation, we give an analytic characterization of elliptic curves in honeycomb form, and we offer a detailed analysis of the tropical group law on such a curve.

Chapter 5 is joint work with Anders Jensen and Elena Rubei and is a departure from the subject of tropical curves. In this chapter, we study tropical determinantal varieties and prevarieties. After recalling the definitions of tropical prevarieties, varieties, and bases, we present a short proof that the 4×4 minors of a 5×n matrix of indeterminates form a tropical basis. The methods are combinatorial and involve a study of arrangements of tropical hyperplanes. Our result together with the results in [DSS05], [Shi10], [Shi11] answer completely the fundamental question of when the r × r minors of a d × n matrix form a tropical basis; see Table 5.1.

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