Branched Galois Covers of Curves: Lifting and Reduction

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This research project concerns algebraic geometry, the study of systems of polynomial equations in many variables. The project focuses on equations formulated in characteristic p for some prime number p. The "characteristic p world" is one where, every time you take p steps forward, you wind up back where you start. Far from being esoteric, such mathematical systems describe the procession of the days of the week (p = 7), the fundamentals of computer architecture (p = 2), and also the setting for important cryptosystems. Algebraic geometry in characteristic p is the study of geometric objects ("varieties") given by solutions to polynomial equations in this setting. This project investigates the relationship between varieties in characteristic p and in characteristic zero ("standard" algebraic geometry), with the goal of shedding light on both worlds. Graduate students will take a major part in the research project. The research will be complemented by educational activities involving the undergraduate math club.The first part of this research project is devoted to understanding when branched Galois covers of curves lift from characteristic p to characteristic zero, and to obtaining information about the geometry and arithmetic of the lifts. In particular, the project aims to obtain a classification of the so-called "local Oort groups" and "weak local Oort groups". The second part of the project involves analyzing integral models of Galois covers of curves over p-adic fields. New tools involving deformation data shed light on stable models of Galois covers of curves with complicated Galois groups. These tools will be exploited further in order to understand stable models of Galois covers as explicitly as possible.