CAREER: Models of curves and non-archimedean geometry

Cuny Baruch College, New York NY

Investigators

Abstract

The "characteristic p world" (for p some prime number) is one where, every time you take p steps forward, you wind up back where you start. Far from being esoteric, this world describes the procession of the days of the week (p = 7), the fundamentals of computer architecture (p = 2), and is also the setting for important cryptosystems. Characteristic p algebraic geometry is the study of geometric objects ("varieties") given by solutions to polynomial equations in this world (e.g., the elliptic curves fundamental to the aforementioned cryptosystems), whereas mixed characteristic algebraic geometry is the study of varieties in a "hybrid" world serving as a bridge from the characteristic p world to our more familiar world where all directions go on forever. A singularity is a place where a variety has crossings, corners, or otherwise fails to be smooth. This project brings new algebraic techniques to bear on the topic of singularities in mixed characteristic algebraic geometry, with one main goal to explicitly analyze what processes are necessary to "resolve" them, i.e, smooth them out. The educational plan for the project provides opportunities at levels from high school to graduate school. Specifically, the PI's graduate students will work directly with the PI on the main research project, the PI will advise undergraduate research in related areas of number theory, the PI will help plan events for Baruch College's chapter of the Association for Women in Mathematics, and the PI will continue his commitment to mathematics education in Liberia by running a mathematics competition for Liberian high school students, as well as a series of remote workshops for their teachers. The first part of the project is dedicated to applications and generalizations of Mac Lane valuations, which give explicit, computationally useful descriptions of normal models of the projective line over a valued field. Mac Lane valuations date back over 80 years, but it is only this past decade that they have been successfully used to attack problems involving models of curves, such as computing resolutions of certain arithmetic surface singularities and verifying conductor-discriminant inequalities. The PI will apply Mac Lane valuations to (1) refine his previous results on conductor-discriminant inequalities for hyperelliptic curves and extend them to superelliptic curves and (2) shed light on the relationship between regular models, semistable models, and the ramification theory of ``purely arithmetic'' covers of Berkovich spaces. Furthermore, he will work to build a theory of Mac Lane valuations for higher genus curves. The second part of the 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 of the space of lifts. The PI will use moduli-theoretic techniques to attack the lifting problem in towers for cyclic covers, and he will continue his long-term work toward a classification of the so-called local Oort groups and the weak local Oort groups. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.