Tropical Geometry of Curves
Algebraic geometry is a classical subject which studies shapes arising as zero sets of polynomial equations. Such objects, called varieties, may be quite complicated but many aspects of their geometry are governed by discrete data. In turn, combinatorial structures arising from particularly meaningful varieties, such as moduli spaces, are interesting in their own right. In recent years tropical geometry has emerged as a robust tool for studying varieties. As a result, many rich connections between algebraic geometry and combinatorics have developed.
Tropical geometry associates polyhedral complexes, like tropical varieties and skeletons, to algebraic varieties. These encode information about the variety or the equations they came from, providing insight in to the underlying combinatorial structure.
In this thesis, I develop tropical geometry of curves from the perspectives of divisors, moduli spaces, computation of skeletons, and enumeration. Already, the realm of curves is rich to explore using the tools of tropical geometry. This thesis is divided into five chapters, each focusing on different aspects of the tropical geometry of curves.
I begin by introducing algebraic curves, tropical curves, and non-Archimedean curves. The ways in which these objects interact will be a common theme throughout the thesis. The picture on the previous page expresses the idea that tropical curves are projections of non-Archimedean curves. Berkovich analytic spaces are heavenly abstract objects which can be viewed by earthly beings through their tropical shadows.
In the second chapter, I develop divisors on tropical curves and tropicalize algebraic divisors. Many constructions for classical curves related to divisors carry over to the tropical world. This will include a tropical Jacobian, a tropical version of the Riemann-Roch theorem, and a tropical Abel-Jacobi map. I first define and compute these objects. Then, I focus on the symmetric power of a curve, because this functions as a moduli space for effective divisors on the curve. I prove that the non-Archimedean skeleton of the symmetric power of a curve is equal to the symmetric power of the non-Archimedean skeleton of the curve. Using this, I prove a realizable version of the tropical Riemann-Roch Theorem.
In the third chapter I focus on moduli spaces. A recurring phenomenon in tropical geometry is that the non-Archimedean skeleton of an algebraic moduli space gives a tropical one. I will develop detailed examples of this. Then, I define a divisorial motivic zeta function for marked stable curves, and prove that it is rational.
In the fourth chapter I compute abstract tropicalizations or non-Archimedean skeletons of a curve. In genus one and two there are known methods for computing these tropicalizations. I develop an algorithm for computing the abstract tropicalizations of hyperelliptic and superelliptic curves. In higher genus, these are the only known results for computing abstract tropicalizations of curves.
In the final chapter I study enumerative problems. Tropical geometry has proven to be a very useful tool in counting curves in the plane. I turn my attention to surfaces in space, and develop tropical counting techniques in this domain. This leads to a preliminary count of binodal cubic surfaces.