Tropical Methods for the Tautological Intersection Theory of the Moduli Spaces of Curves

曲线模空间同义反复交集理论的热带方法

• 批准号: 2100962
• 负责人: Renzo Cavalieri
• 金额: $ 16.5万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100962&HistoricalAwards=false
• 关键词: Tropical; Methods; Tautological; Intersection; Theory

Algebraic geometry is a broad and active area of research in mathematics. Moduli spaces, geometric objects whose points parameterize other geometric objects, are of fundamental importance both in algebraic geometry, and in connecting algebraic geometry to other areas of science. For example, the connection with physics arises from the fact that the evolution of strings in space-time may be interpreted as an appropriate measurement on a moduli space of stable maps to space-time. The geometry of moduli space is extremely sophisticated, but it often comes with a rich recursive structure: in simple terms, more complicated moduli spaces contain within themselves a skeleton built of simpler moduli spaces. Over the last few decades this phenomenon has led to the development of several combinatorial approaches to the study of intersection theory of moduli spaces. The main goal of this project is to develop a thorough understanding of the intersection theory of a particular class of moduli spaces, called admissible cover spaces. Admissible cover spaces provide a rich and interesting connection between algebraic geometry and representation theory of finite groups, and have significant applications to mathematical physics and mirror symmetry. The goal will be achieved through a combination of several techniques and perspectives, including methods coming from tropical geometry, logarithmic geometry and mathematical physics. The PI, together with collaborators, will work in parallel both to further develop and to apply these techniques to the study of the structure of moduli spaces of admissible covers. This project provides research training opportunities for students.Specific projects contributing to achieving the main goal are organized in three groups. The first group of projects explores the structure of classes of hyperelliptic curves with marked Weierstrass points and pairs of conjugate points. The aim is to generalize the notion of Cohomological Field Theory, and to exploit this structure to obtain graph formulas for these classes. A better understanding of the structure of admissible cover loci is a tool to recover enumerative information hidden in Gromov-Witten invariants of curves. The second group of projects aims to give a solid combinatorial framework for the tautological intersection theory of a directed system of birational models of the moduli space of curves, obtained by blowing up all boundary strata (and proper transforms thereof). Besides being of independent interest, we expect this calculus to be an important tool in understanding families of classes of admissible covers, whose intersection with the boundary is transversalized in these birational transforms. These techniques allow new perspectives on the computation of double Hurwitz numbers, and the generalization to similar enumerative geometric problems on moduli spaces of twisted log-canonical divisors. Tropical geometry plays a fundamental role in organizing the birational modifications of the moduli space of curves that we expect to use in the study of cycles of admissible covers. The last group of projects builds on foundational work in tropical geometry that the PI conducted in collaboration with Gross and Markwig. Having defined a theory of tropical psi classes, the goal is now to establish a rigorous tropicalization statement relating algebraic and tropical classes, with the expectation that these will play a significant role in connecting algebraic and tropical intersection theory.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.

代数几何形状是数学研究的广泛而活跃的领域。模量空间，几何对象，其要点化其他几何对象的点在代数几何形状以及将代数几何形状连接到其他科学领域都至关重要。例如，与物理学的联系源于以下事实：在时空中，字符串的演变可以解释为在稳定地图到时空的模量空间上的适当测量。模量空间的几何形状非常复杂，但通常具有丰富的递归结构：简单地说，更复杂的模量空间内部包含一个由更简单的模量空间构建的骨架。在过去的几十年中，这种现象导致了多种组合方法研究模量空间相交理论的发展。该项目的主要目标是对特定类别的模量空间的交叉理论（称为可允许的覆盖空间）进行彻底理解。可接受的覆盖空间在代数几何形状和有限基团的表示理论之间提供了丰富而有趣的联系，并在数学物理和镜像对称性方面具有重要的应用。该目标将通过多种技术和观点的结合来实现，包括来自热带几何形状，对数几何形状和数学物理学的方法。 PI与合作者一起将同时使用，以进一步开发并将这些技术应用于可允许覆盖的模量空间结构的研究。该项目为学生提供了研究培训机会。为实现主要目标做出贡献的特定项目分为三组。第一组项目探索了具有明显的Weierstrass点和成对共轭点的过椭圆曲线类别的结构。其目的是概括共同体田地理论的概念，并利用这种结构来获得这些类别的图公式。更好地理解可允许的封面基因座的结构是一种恢复隐藏在Gromov-witten曲线中的枚举信息的工具。第二组项目旨在为曲线模量模型空间的二元模型的有向模型的重言式相交理论提供一个可靠的组合框架，该系统通过炸毁了所有边界层（及其适当的变换）而获得。除了具有独立的兴趣外，我们还希望这种演算是理解可允许的覆盖范围家庭的重要工具，该涵盖范围的家庭与边界的相交在这些异性转变中被横视了。这些技术允许对双Hurwitz数字计算的新观点，以及对扭曲的对数典型除数模量空间上类似列举几何问题的概括。热带几何形状在组织曲线模量空间的生育修饰方面起着至关重要的作用，我们期望将其用于研究可允许覆盖的循环。最后一组项目基于PI与Gross和Markwig合作进行的热带几何形状的基础工作。在定义了热带PSI类理论之后，目标现在是建立与代数和热带类别有关的严格的热带化声明，并期望这些声明将在连接代数和热带交集理论中起重要作用。使用基金会的知识分子优点和更广泛的审查标准，通过评估被认为值得支持。