Combinatorial and Tropical Degenerations of del Pezzo Surfaces and Their Moduli

热带几何是数学中一个年轻而迅速发展的领域，植根于代数几何、复分析、交换代数和组合学，除其他数学领域外，还应用于计算机科学、生物学和统计物理。近十年来，这门学科取得了巨大的发展，既确立了这一领域的独立地位，又揭示了它与众多纯数学和应用数学分支的深刻联系。本研究项目将为经典模空间族，即del Pezzo曲面，提供一个新的组合视角。这个方向上的计算挑战是它的驱动力。主要目标是加深对这些对象的理解，并通过热带几何中的具体计算开发新的技术来分析这些抽象空间。新的发现将有助于快速增长的开源数学软件Sage。在这个项目中调查的几个主题见证了学科的跨学科性质，适合与研究生合作研究。热带几何为使用具体的组合工具解决代数几何问题提供了一个框架：代数变体被加权的、平衡的多面体复合物所取代。这些对象只保留了足够的原始变量数据以保持其意义，同时丢弃了它们的大部分复杂性。它们的组合在很大程度上依赖于我们变异的嵌入。本研究的目的是从退化几何、热带几何和非阿基米德几何的角度，发展有效的组合方法来研究del Pezzo曲面和相关几何对象的模空间。该项目有三个组成部分。首先，通过多面体组合建立del Pezzo曲面及其低阶模的忠实热带化和组合特征化的有效方法，以利用几何和拓扑不变量。第二，解释在热带环境下低度del Pezzo曲面上著名的经典枚举几何结果。最后，PI将应用多面体技术建立新的障碍，以将曲线从热带提升到代数世界，解决热带曲线的“过剩”问题，并探索涉及del Pezzo曲面上各种已知坐标系之间的非理性和过渡图的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Tropical geometry is a young and rapidly growing area in mathematics, rooted in algebraic geometry, complex analysis, commutative algebra, and combinatorics, with applications in computer science, biology, and statistical physics, in addition to other areas of mathematics. The recent decade has seen tremendous development in the subject that both established the field as an area in its own right and unveiled its deep connections to numerous branches of pure and applied mathematics. This research project will provide a new combinatorial perspective on a classical family of moduli spaces, namely del Pezzo surfaces. Computational challenges in this direction are its driving force. The primary goal is to deepen understanding of these objects and develop new techniques to analyze such abstract spaces through concrete computations in tropical geometry. New findings will contribute to the rapidly growing open source mathematical software Sage. Several topics investigated in this project witness the interdisciplinary nature of the subject and are suitable for research in collaboration with graduate students.Tropical geometry provides a framework for solving algebro-geometric problems using concrete combinatorial tools: algebraic varieties are replaced by weighted, balanced polyhedral complexes. These objects preserve just enough data about the original varieties to remain meaningful, while discarding much of their complexity. Their combinatorics depends strongly on the embeddings of our varieties. The objective of this research is to develop effective combinatorial methods to study moduli space of del Pezzo surfaces and related geometric objects from the perspective of degenerations, tropical and nonarchimedean geometry. The project has three components. First, to establish effective methods for constructing faithful tropicalization and characterizing the combinatorics of del Pezzo surfaces and their moduli in low degrees to harness geometric and topological invariants via polyhedral combinatorics. Second, to interpret well-known classical enumerative geometry results on del Pezzo surfaces of low degrees in the tropical setting. Finally, the PI will apply polyhedral techniques to establish new obstructions to lifting curves from the tropical to the algebraic world, addressing "superabundance" of tropical curves, and explore questions involving unirationality and transition maps between various known coordinate systems on del Pezzo surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.