Descent, rational points, and the geometry of moduli spaces

丢番图几何是通过复数解的几何棱镜来研究多项式方程的整数解。 整数解的结构特征有时受微妙的几何现象控制。 我们依赖于复杂表面几何形状的结果，特别是那些由小次方程定义的表面。 示例包括三个变量的三阶或四阶方程。 我们的希望是辨别管理具有共同特征的大类问题的行为的更大模式。该项目解决了代数和丢番图几何界面上的问题，这些问题是由有关代数簇上有理点的行为的基本问题引起的。 在最简单的情况下，这些都涉及到经典代数几何的美丽结构。除此之外，人们很快就会遇到通过经典技术无法解决的深刻几何问题。 具体的研究问题包括：如何从几何角度解释具有水平结构的K3曲面的模空间？使用 K3 曲面及其扭曲类似物的派生等价概念可以在多大程度上组织这些内容？这些技术可以用来明确评估布劳尔-马宁障碍吗？给定两个导出的等效 K3 表面，它们的丢番图性质如何相关，尤其是在局部场上？对于曲线上的 del Pezzo 纤维，有理曲线空间如何受上同调不变量控制？随着纤维化的数值不变量的变化，有理曲线空间显示出什么归纳结构？ 这些问题将通过有理曲线的变形理论特性、布里奇兰稳定条件产生的有效曲线锥体的结构描述以及 K3 纤维简并纤维的分类来解决。

Diophantine geometry is the study of integer solutions of polynomial equations, seen through the prism of the geometry of their solutions over the complex numbers. Structural characteristics of the integer solutions are sometimes governed by subtle geometric phenomena. We rely on results on the geometry of complex surfaces, especially those defined by equations of small degree. Examples include equations of degree three or four in three variables. Our hope is to discern larger patterns governing the behavior of large classes of problems sharing common characteristics.This project addresses problems at the interface of algebraic and Diophantine geometry arising from fundamental questions about the behavior of rational points on algebraic varieties. In the simplest situations, these touch on beautiful constructions from classical algebraic geometry. Beyond these, one quickly encounters deep geometric problems not accessible through classical techniques. Specific research questions will include: How to interpret moduli spaces of K3 surfaces with level structure geometrically? To what extent can these be organized using notions of derived equivalence for K3 surfaces and their twisted analogs? Can these techniques be used to evaluate Brauer-Manin obstructions explicitly? Given two derived-equivalent K3 surfaces, how are their Diophantine properties related, especially over local fields? For del Pezzo fibrations over curves, how are spaces of rational curves governed by cohomological invariants? As the numerical invariants of the fibrations vary, what inductive structures are shown by the spaces of rational curves? These will be addressed using deformation-theoretic properties of rational curves, structural descriptions of cones of effective curves arising from Bridgeland stability conditions, and classifications of degenerate fibers of K3 fibrations.