Combinatorial and Tropical Degenerations of del Pezzo Surfaces and Their Moduli

del Pezzo 表面的组合和热带退化及其模量

• 批准号: 1954163
• 负责人: Maria Cueto
• 金额: $ 15万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954163&HistoricalAwards=false
• 关键词: Combinatorial; Tropical; Degenerations; del; Pezzo

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.

热带几何是数学中一个年轻而迅速发展的领域，它植根于代数几何、复分析、交换代数和组合学，除了数学的其他领域外，还应用于计算机科学、生物学和统计物理。近十年来，这门学科有了巨大的发展，既建立了该领域作为一个独立的领域，又揭示了它与纯数学和应用数学的许多分支的深刻联系。这项研究将为一类经典的模空间，即del Pezzo曲面，提供一个新的组合视角。这个方向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，并开发新的技术，通过热带几何中的具体计算来分析这种抽象的空间。新的发现将有助于迅速发展的开源数学软件Sage。这个项目中研究的几个主题证明了这门学科的跨学科性质，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：代数族被加权、平衡的多面体复合体取代。这些对象保留了关于原始变种的足够数据，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，从退化几何、热带几何和非阿基米德几何的角度研究del Pezzo曲面及其相关几何对象的模空间。该项目有三个组成部分。首先，通过多面体组合数学建立有效的方法来构造忠实的热带化并刻画Del Pezzo曲面及其模的低次组合，从而利用几何和拓扑不变量。第二，解释热带环境下低次Del Pezzo曲面上著名的经典计数几何结果。最后，PI将应用多面体技术来建立新的障碍，将曲线从热带提升到代数世界，解决热带曲线的“超丰度”问题，并探索涉及Del Pezzo表面各种已知坐标系之间的单旋性和过渡图的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Combinatorics and Real Lifts of Bitangents to Tropical Quartic Curves

热带四次曲线双切线的组合学和实升力

• DOI: 10.1007/s00454-022-00445-1
• 发表时间: 2023
• 期刊: Discrete & Computational Geometry
• 影响因子: 0.8
• 作者: Cueto, Maria Angelica;Markwig, Hannah
• 通讯作者: Markwig, Hannah