Tropical geometry and the moduli space of Prym varieties

热带几何和 Prym 簇的模空间

• 批准号: EP/X002004/1
• 负责人: Yoav Len
• 金额: $ 44.68万
• 依托单位: University of St Andrews
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX002004%2F1
• 关键词: Tropical; geometry; moduli; space; Prym

Algebraic geometry is concerned with geometric objects that arise as solutions to polynomial equations. Such objects, known as algebraic varieties, are at the heart of many real-world problems and have been studied since the dawn of maths. In recent decades, it was observed that significant mileage may be gained by stripping away some of the geometry and focusing instead on combinatorial aspects of those object. This pivotal shift in perspective is the basis for tropical geometry and has recently led to major breakthroughs in the geometry of curves, enumerative geometry, combinatorics, mathematical physics, and number theory. The current proposal is concerned with a family of important algebraic groups known as Prym varieties, which play a key role in rationality questions for threefolds, construction of compact hyper-Kähler manifolds , and the birational geometry of the moduli of abelian varieties. Despite numerous attempts by various authors, it is not yet known how to fully construct a compact space classifying them, and what the structure of such a space would be. However, by appealing to tropical tools such as the recently discovered tropical Prym variety, the time is ripe to study Prym varieties through a combination of algebraic, non-archimedean, logarithmic, and combinatorial techniques

代数几何形状与作为多项式方程的解决方案产生的几何对象有关。这种称为代数品种的物体是许多实际问题的核心，自数学曙光以来就已经进行了研究。近几十年来，观察到，可以通过剥离一些几何形状并专注于这些物体的组合方面来获得重要的里程碑。这种观点的关键转变是热带几何形状的基础，最近导致了曲线，枚举几何形状，组合学，数学物理学和数字理论的几何形状的重大突破。当前的提案与一个称为Prym品种的重要代数群的家族有关，该家族在三倍的合理性问题中起关键作用，紧凑的Hyper-Kähler歧管的构建以及Abelian品种模量的亿万富翁几何形状。尽管各种作者进行了许多尝试，但尚不知道如何完全构造它们对它们进行分类的紧凑空间，以及这种空间的结构是什么。然而，通过似乎是热带工具，例如最近发现的热带Prym品种，通过代数，非Archimedian，对数和组合技术的组合来研究PRYM变化的时间已经成熟。