Descent, rational points, and the geometry of moduli spaces

丢番图几何是研究多项式方程的整数解，通过其解在复数上的几何棱镜来观察的。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，特别是那些由小次数方程定义的结果。例如三个变量的三次或四次方程。我们希望找出更大的模式来管理具有共同特征的大类问题的行为。这个项目解决了代数几何和丢番图几何交界处的问题，这些问题是由关于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层次几何问题。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？在多大程度上可以使用K3曲面及其扭曲类似物的导出等价性概念来组织这些曲面？这些技术能否用于明确评估Brauer-Manin梗阻？给出两个导数等价的K3曲面，它们的丢番图性质有什么关系，特别是在局部域上？对于曲线上的del Pezzo纤维，有理曲线的空间是如何由上同调不变量支配的？随着有理曲线的数值不变量的变化，有理曲线的空间表示出什么样的归纳结构？这些问题将通过有理曲线的变形理论性质、由Bridgeland稳定性条件产生的有效曲线圆锥的结构描述以及K3纤维的退化纤维分类来解决。

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.