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

在本质上，算术几何是一门旨在理解多变量多项式方程组的学科，其约束条件是解的坐标是有理数或整数。 多项式方程组有一个几何化身，称为代数簇。 算术几何的咒语是代数簇的几何性质，如曲率，强烈地影响着多项式方程组的解的类型和结构。 PI研究多项式方程的系统，其代数簇是曲面，即，它们是二维的。 代数曲面分为四个粗糙门。在这个分类中，PI研究了K3曲面类，属于“中等复杂性”门。“我们对K3曲面几何的知识已经足够成熟，可以对它们所包含的合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，是否有一种算法可以确定方程组是否有解？ 如果是这样的话，这个算法可以实现吗？K3曲面具有丰富且结构化的几何形状;它们的算法是微妙的，但可以想象的是易于处理的。在过去的10年里，大量的研究成果都指向一个令人惊讶的期望：给定数域上的K3曲面方程，应该有一种算法来检测曲面是否存在有理点。许多项目的目的是提供证据，实现这一期望，或提供实用的方法来检测存在的合理点的K3曲面。 建议的项目包括详细研究具有Brauer群层次结构的K3曲面的模空间，以及研究K3曲面的可能的代数Brauer群的一类逆Galois问题。 PI还建议研究代数簇上的一类在有理点和整点之间插值的轨道点，并以统一的方式进行研究，以便更好地了解何时必须将这两种类型的点分开处理。该奖项反映了NSF的法定使命，并通过利用基金会的知识价值和更广泛的影响进行评估，被认为值得支持审查标准。

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.