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，通过其解决方案在复数的几何棱镜看到。 整数解的结构特征有时受微妙的几何现象所支配。 我们依赖于复杂曲面的几何形状的结果，特别是那些定义的小度方程。 例子包括三个变量的三次或四次方程。 我们的希望是识别更大的模式管理的行为大类的问题共享common characteristics.This项目解决的问题，在接口的代数和丢番图几何所产生的基本问题的行为合理的点代数簇。 在最简单的情况下，这些触及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层几何问题。 具体的研究问题包括：如何几何地解释具有层次结构的K3曲面的模空间？在何种程度上可以组织这些使用的概念，导出等价的K3曲面及其扭曲的类似物？这些技术可以用来明确评估Brauer-Manin障碍吗？给定两个导出等价的K3曲面，它们的丢番图性质是如何联系的，特别是在局部域上？对于曲线上的del Pezzo纤维化，有理曲线的空间如何受上同调不变量的支配？当纤维化的数值不变量变化时，有理曲线的空间显示了什么样的归纳结构？ 这些将被解决使用变形理论的合理曲线，结构描述的有效曲线的锥所产生的Bridgeland稳定条件，和退化纤维的K3纤维化的分类。

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.