AI 中文总结
该研究在层背景下阐述群块定理，开发代数几何类似物取代模型理论工具，通过研究任意范畴中部分态射等技术工作，扩展已知结果，还在模型理论和代数几何方面分别发展相关理论。
AI 中文摘要
我们在层的背景下阐述了一个群块定理，它推广了模型理论和代数几何中的许多类似结果。其次，我们开发了一种代数几何类似物，用于从平稳类型上可定义函数的芽产生群块的赫鲁绍夫斯基方法。用希尔伯特概型研究“典范”有理态射族取代了典范基和虚元消除的模型理论工具，使我们能够在更一般的基概型上扩展先前已知的结果。这些结果的证明涉及一些可能具有独立意义的技术工作。首先，我们研究任意范畴中的部分态射，并表明小范畴上的部分 magma 预层允许到群的泛态射。在模型理论方面，我们表明\(M^{eq}\)的型可定义集可以解释为型可定义集的层商。在代数几何方面，我们发展了任意基上概型的有理态射和有理态射族的理论。
英文摘要
We formulate a group chunk theorem in the context of sheaves on sites which generalizes many similar results in model theory and algebraic geometry. Secondly, we develop an algebro-geometric analogue of Hrushovski's method of producing a group chunk from germs of definable functions on stationary types. The use of the model-theoretic tools of canonical bases and elimination of imaginaries is replaced with the use of Hilbert schemes to study ``canonical'' families of rational morphisms, allowing us to extend the previously known results over more general base schemes. The proofs of these results involve some technical work which may be of independent interest. First, we study partial morphisms in arbitrary categories, and show that a presheaf of partial magmas on a small category admits a universal morphism to a group. On the model theory side, we show that type-definable sets of $M^{eq}$ can be interpreted as sheaf quotients of type-definable sets. On the algebraic geometry side, we develop a theory of rational morphisms and families of rational morphisms of schemes over an arbitrary base.
CommentsPhD Thesis, University of California Berkeley, 2026. Advisors: Thomas Scanlon and Martin Olsson. 124 pages
