Level Structures on K3 Surfaces, and Constrained Rational Points on Log Fano Varieties

William Marsh Rice University, Houston TX

Investigators

Abstract

At heart, arithmetic geometry is a subject that aims to understand systems of polynomial equations in many variables, with the constraint that the coordinates of the solutions be rational numbers or integers. A system of polynomial equations has a geometric avatar, called an algebraic variety. The mantra of arithmetic geometry is that geometric properties of an algebraic variety, like curvature, bear strongly on the types and the structure of solutions to the underlying system of polynomial equations. The PI studies systems of polynomial equations whose algebraic varieties are surfaces, i.e., they are two-dimensional. Algebraic surfaces are classified into four rough phyla. Within this classification, the PI studies the class of K3 surfaces, within a phylum of "intermediate complexity." Our knowledge of the geometry of K3 surfaces is ripe enough now for a meaningful study of the rational points they harbor. The projects supported by this award aim to answer questions like: given a set of polynomial equation that define a K3 surface, is there an algorithm that will determine whether the system of equations has solutions? If so, can this algorithm be implemented? K3 surfaces have a rich and structured geometry; their arithmetic is subtle, yet conceivably tractable. A host of conjectures in the last 10 years point to a surprising expectation: given equations for a K3 surface over a number field, there should be an algorithm that detects whether the surface caries rational points. Many of the projects are designed to either provide evidence towards this expectation, or to give practical methods for detecting the existence of rational points on K3 surfaces. Proposed projects include a detailed study of moduli spaces of K3 surfaces with Brauer group level structures, as well as a type of inverse Galois problem to study the possible algebraic Brauer groups of K3 surfaces. The PI also proposes to study a class of orbifold points on algebraic varieties that interpolate between rational points and integral points, and to do so in a uniform way, so as to develop a better understanding of when treatments of these two types of points must be separated. 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.