Combinatorial and Tropical Degenerations of Classical Moduli Spaces

经典模空间的组合和热带退化

• 批准号: 1700194
• 负责人: Maria Cueto
• 金额: $ 13.5万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700194&HistoricalAwards=false
• 关键词: Combinatorial; Tropical; Degenerations; Classical; Moduli

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. Moduli spaces are spaces of parameters that classify algebro-geometric mathematical objects, such as varieties, maps between varieties, and vector bundles. This research project aims at providing a new combinatorial perspective on classical moduli spaces and their interplay with nonarchimedean analytic geometry. 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. 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. By contrast, the Berkovich space or the space of valuations is independent of all such choices. By choosing appropriate coordinates, the geometry and topology of these valuations are reflected in rich combinatorial properties on the polyhedral side, including connectedness, shellability, etc. Thus, the intricacies of the original space turn into exciting combinatorial problems. Tropicalizations resulting from these nice embeddings are said to be faithful. The project aims at developing certificates for local faithfulness by means of initial (Groebner) degenerations of the variety. The notion of faithfulness near that point then turns into desirable properties of this degeneration. Not surprisingly, validating these properties requires a deep understanding of the combinatorics and geometry of the initial degenerations. Thus, the need to find effective methods for constructing or detecting faithfulness is at the core of tropical geometry. The project will develop such program for several important classes of examples: Grassmannians, moduli of curves, del Pezzo surfaces, Fano schemes and cluster varieties.

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。模空间是对代数几何数学对象进行分类的参数空间，例如变量、变量之间的映射和向量束。本研究项目旨在为经典模空间及其与非阿基米德解析几何的相互作用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。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。相比之下，Berkovich空间或估值空间与所有这些选择无关。通过选择合适的坐标，这些估值的几何和拓扑结构反映在多面体侧丰富的组合属性中，包括连通性、可壳性等。因此，原始空间的复杂性变成了令人兴奋的组合问题。由这些漂亮的嵌入产生的热带化被认为是忠实的。该项目旨在通过品种的初始（格罗布纳）退化来开发本地忠诚证书。在这一点附近，忠诚的概念变成了这种退化的理想属性。毫不奇怪，验证这些性质需要对初始退化的组合学和几何有深刻的理解。因此，需要找到有效的方法来构建或检测忠实度是热带几何的核心。该项目将为几个重要的例子类开发这样的程序：Grassmannians、曲线模、del Pezzo曲面、Fano方案和簇品种。

项目成果

Initial degenerations of Grassmannians

格拉斯曼人的最初退化

• DOI: 10.1007/s00029-021-00679-6
• 发表时间: 2021
• 期刊: Selecta Mathematica
• 作者: Corey, Daniel

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

Tropical geometry of genus two curves

属两条曲线的热带几何

• DOI: 10.1016/j.jalgebra.2018.08.034
• 发表时间: 2019
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Cueto, Maria Angelica;Markwig, Hannah
• 通讯作者: Markwig, Hannah