Descent, rational points, and the geometry of moduli spaces

丢番图几何学是对多项式方程的整数解的研究，通过复数解的几何棱镜来观察。整数解的结构特征有时受微妙的几何现象支配。我们依赖于复杂曲面的几何结果，特别是那些由小次方程定义的曲面。例子包括三个变量的三次或四次方程。我们的希望是发现更大的模式，这些模式控制着具有共同特征的大类问题的行为。这个项目解决了代数和丢番图几何的接口问题，这些问题是由代数变量上有理点的行为的基本问题引起的。在最简单的情况下，它们触及经典代数几何中的美丽结构。除此之外，人们很快就会遇到经典技术无法解决的深刻几何问题。具体的研究问题将包括：如何从几何上解释具有水平结构的K3曲面的模空间？在多大程度上，这些可以用K3表面及其扭曲类似物的推导等效概念来组织？这些技术可以用来明确地评估布劳尔-马宁障碍吗？给定两个等效的K3曲面，它们的丢番图性质是如何相关的，特别是在局部场上？对于曲线上的del Pezzo振动，如何用上同调不变量控制有理曲线的空间？当振动的数值不变量变化时，在有理曲线的空间中表现出什么样的归纳结构？这些问题将通过合理曲线的变形理论性质、由布里奇兰稳定条件产生的有效曲线锥的结构描述以及K3纤维的退化分类来解决。

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.