Combinatorial and nonarchimedean methods in algebraic geometry

研究者将在代数几何方面进行几项研究，包括组合和非阿基米德方法的应用，以研究代数曲线及其模，以及交集理论。特别地，本文讨论了Gieseker-Petri定理和极大秩猜想的非阿基米德方法，曲线模的上同调上的权值过滤，热带化的度量性质和分析，以及交集的泛函热带化理论的发展。代数几何研究多项式方程组的解集。在非阿基米德领域，人们可以把理解这样一个解集的问题分成两个部分。解的可能值是多少？给定估值的解是什么？解的赋值集具有丰富的组合和多面体结构，是热带几何研究的主要对象。该领域的最新发展使得利用这些估值集的几何来解决有关实际解集几何的微妙问题成为可能。目前的提案旨在完善这些新方法，并探索在代数几何开放问题的更深层次的应用。

The investigator will pursue several lines of research in algebraic geometry involving the application of combinatorial and nonarchimedean methods to study algebraic curves and their moduli, plus intersection theory. In particular, this proposal deals with nonarchimedean approaches to the Gieseker-Petri Theorem and Maximal Rank Conjectures, the weight filtration on cohomology of moduli of curves, metric properties of tropicalizations and analytifications, and the development of a functorial tropicalization of intersection theory.Algebraic geometry studies solution sets of systems of polynomial equations. Over a nonarchimedean field, one can split the problem of understanding such a solution set into two parts. What are the possible valuations of solutions? And what are the solutions with a given valuation? The set of valuations of solutions has a rich combinatorial and polyhedral structure, and is the primary object of study in tropical geometry. Recent developments in this field make it possible to resolve subtle questions about the geometry of the actual solution set using the geometry of these sets of valuations. The current proposal aims to refine these new methods and explore deeper applications to open problems in algebraic geometry.