Combinatorial and Tropical Degenerations of del Pezzo Surfaces and Their Moduli

热带几何是数学中一个年轻且快速发展的领域，它植根于代数几何、复分析、交换代数和组合数学，在计算机科学、生物学和统计物理学以及其他数学领域都有应用。近十年来，这一学科取得了巨大的发展，既确立了这一领域本身的地位，又揭示了它与纯数学和应用数学众多分支的深刻联系。这个研究计画将提供一个新的组合观点在一个经典的家庭的模空间，即德尔佩佐表面。在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物体的理解，并开发新技术，通过热带几何中的具体计算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。热带几何提供了一个框架，使用具体的组合工具解决代数几何问题：代数簇被取代的加权，平衡多面体复形。这些对象只保留了足够的关于原始变种的数据，以保持有意义，同时丢弃了它们的大部分复杂性。它们的组合学很大程度上依赖于我们的变种的嵌入。本研究的目的是发展有效的组合方法来研究模空间的del Pezzo曲面和相关的几何对象的退化，热带和nonarchimedean几何的角度。该项目有三个组成部分。首先，建立有效的方法来构建忠实的热带化和表征del Pezzo曲面及其低次模的组合，通过多面体组合来利用几何和拓扑不变量。第二，解释著名的经典枚举几何的结果del Pezzo曲面的低度在热带设置。最后，PI将应用多面体技术建立新的障碍，以提高曲线从热带到代数世界，解决“过剩”的热带曲线，该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估，更广泛的影响审查标准。

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.