Computations and Moduli Spaces for Non-archimedean Varieties
Tropical geometry and non-archimedean analytic geometry study algebraic varieties over a field K with a non-archimedean valuation. One of the major goals is to classify varietiess over K by intrinsic tropical properties. This thesis will contain my work at UC Berkeley and my joint work with others towards the goal.
Chapter 2 discusses some moduli spaces and their tropicalizations. The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.
Chapter 3 develops numerical algorithms for Mumford curves over the field of p-adic numbers. Mumford curves are foundational to subjects dealing with non-archimedean varieties, and it has various applications in number theory. We implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings.
Chapter 4 discusses how to tropicalize del Pezzo surfaces of degree 5, 4 and 3. A generic cubic surface P^3 is a Del Pezzo surface of degree 3, which is obtained by blowing up the plane at 6 points. We study its tropicalization by taking the intrinsic embedding of the surface surface minus its 27 lines. Our techniques range from controlled modifications to running gfan on the universal Cox ideal over the relevant moduli space. We classify cubic surfaces by combinatorial properties of the arrangement of 27 trees obtained from the image of the 27 lines under this tropicalization.
Chapter 5 discusses the classical Cayley-Bacharach theorem, which states that if two cubic curves on the plane intersect at 9 points, then the 9th point is uniquely determined if 8 of the points are given. The chapter derives a formula for the coordinates of the 9th point in terms of the coordinates of the 8 given points. Furthermore, I will discuss the geometric meaning of the formula, and how it is related to del Pezzo surfaces of degree 3.