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

代数几何是数学中一个广泛而活跃的研究领域。模空间是几何对象，其点可参数化其他几何对象，在代数几何中以及在将代数几何与科学的其他领域联系起来时，模空间都是基本重要的。例如，与物理学的联系源于这样一个事实，即弦在时空中的演化可以被解释为稳定映射到时空的模空间上的适当度量。模空间的几何非常复杂，但它通常带有丰富的递归结构：简单地说，更复杂的模空间本身包含由更简单的模空间构建的骨架。在过去的几十年里，这一现象导致了几种组合方法的发展，以研究模空间的交理论。这个项目的主要目的是对一类特殊的模空间的交集理论有一个透彻的理解，这类模空间被称为允许覆盖空间。可容许覆盖空间在代数几何和有限群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方面有重要的应用。这一目标将通过几种技术和观点的结合来实现，包括来自热带几何、对数几何和数学物理的方法。PI将与合作者一起并行工作，进一步发展和应用这些技术来研究可容许覆盖的模空间的结构。该项目为学生提供研究培训机会。有助于实现主要目标的具体项目分三组进行。第一组项目探索具有标记的魏尔斯特拉斯点和共轭点对的超椭圆曲线类的结构。其目的是推广上同调场理论的概念，并利用这种结构来获得这些类的图形公式。更好地理解允许覆盖轨迹的结构是恢复隐藏在曲线的Gromov-Witten不变量中的计数信息的工具。第二组项目旨在为重言式交集理论提供一个坚实的组合框架，该理论是通过炸毁所有边界层(及其适当的变换)而获得的曲线模空间的二元模型的定向系统。除了具有独立的意义外，我们还期望这一演算成为理解可容许覆盖族的一个重要工具，这些覆盖族与边界的交点在这些双胞胎变换中被转换。这些技巧为计算双Hurwitz数和推广到扭曲对数正则因子的模空间上的类似计数几何问题提供了新的视角。热带几何在组织曲线的模空间的双态修改方面起着重要的作用，我们期望用它来研究可容许覆盖的循环。最后一组项目建立在热带几何基础工作的基础上，这些工作是由PI与Gross和Markwig合作进行的。在定义了热带psi类的理论之后，现在的目标是建立一个严格的热带化声明，将代数类和热带类联系起来，期望这些将在连接代数和热带交叉理论方面发挥重要作用。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.