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

从本质上讲，算术几何是一门旨在理解多变量多项式方程系统的学科，其解的坐标必须是有理数或整数。多项式方程组有一个几何化身，称为代数变体。算术几何的口头禅是，一个代数变量的几何性质，比如曲率，对多项式方程底层系统的解的类型和结构有着强烈的影响。PI研究的是多项式方程组，它们的代数变体是曲面，即它们是二维的。代数曲面被分为四个粗糙的门类。在这种分类中，PI研究K3表面的类别，在“中等复杂性”的门中。我们对K3曲面的几何知识现在已经足够成熟，可以对它们所包含的有理点进行有意义的研究。该奖项支持的项目旨在回答以下问题：给定一组定义K3曲面的多项式方程，是否存在一种算法来确定方程组是否有解？如果可以，这个算法可以实现吗？K3表面具有丰富和结构化的几何形状；他们的计算很微妙，但可以想象的是易于处理。在过去的10年里，大量的猜想指向了一个令人惊讶的期望：给定一个K3曲面在一个数字域上的方程，应该有一个算法来检测该曲面是否含有有理点。许多项目的目的要么是为这种期望提供证据，要么是提供检测K3表面上有理点存在的实用方法。提出的项目包括详细研究具有Brauer群水平结构的K3曲面的模空间，以及研究K3曲面可能的代数Brauer群的一类逆伽罗瓦问题。PI还提出研究一类在有理点和积分点之间插值的代数变量上的轨道点，并以统一的方式进行插值，以便更好地理解这两类点的处理何时必须分开。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.