On the Motivic Goettsche Invariants

代数几何是研究多项式方程的解的学科，在数学、物理学和工程学的其他领域也有许多应用。在代数几何中，一些研究得很好的对象是射影平面（在无穷远处有额外的点的平面）和其上的曲线。一个自然的问题是找到满足某些特殊条件的曲线的数量，例如通过不动点或具有特殊形状，这是代数几何的一个分支的起点，称为枚举几何。在过去的20年里，物理学中的枚举几何和弦理论之间的相互作用产生了许多令人兴奋的新发展，并且一直是当前研究的中心领域。 这个研究项目的目的是定义和研究新的不变量，在更广泛的意义上一般代数曲面，有连接到数学和物理。这些不变量包含关于奇异曲线的数目、曲面的几何性质以及奇异曲线的参数化空间的复杂信息。该项目设想开发一个理论框架和计算工具，并在数学的不同学科之间建立联系。 这将增加对代数簇几何的理解，并可应用于弦理论中时空的描述。本研究项目处理代数几何中的几个问题，特别是关于代数曲面上奇异曲线的计数及其motivic推广。以前的工作已经建立了具有某些给定奇点的簇的个数满足普适公式。这个项目涉及这些普遍公式在动机意义上的推广。本研究定义了一系列的motivic不变量，并研究了它们的数量特性以及它们的几何解释。这项工作的目的是扩展与Gromov-Witten理论，稳定对理论和这些新的不变量的模形式的现有连接，并将它们与其他motivic曲线计数不变量进行比较。这些新的不变量的优点之一是它们满足很好的公式，并且只依赖于拓扑交数。

Algebraic geometry is the study of solutions of polynomial equations, with many applications in other parts of mathematics, physics, and engineering. Some of the well-studied objects in algebraic geometry are the projective plane (the plane with additional "points at infinity") and curves on it. A natural question is that of finding the number of curves that satisfy certain special conditions, such as passing through fixed points or having special shapes, which was the starting point of a branch of algebraic geometry called enumerative geometry. During the past twenty years, the interaction between enumerative geometry and string theory in physics has resulted in many exciting new developments and has been a central area of current research. This research project aims to define and study new invariants, in a broader sense on general algebraic surfaces, that have connections to both mathematics and physics. These invariants contain sophisticated information about the number of singular curves, the geometric properties of the surface, and the parametrization space of singular curves. The project envisages development of a theoretical framework and tools for computation, as well as establishing connections between different subjects in mathematics. It will add to the understanding of geometry of algebraic varieties and has application to the description of spacetime in string theory.This research project treats several problems in algebraic geometry, especially about the enumeration of singular curves on algebraic surfaces and its motivic generalization. Previous work has established that the numbers of varieties with certain given singularities satisfy universal formulas. This project concerns the generalization of these universal formulas in the motivic sense. The research defines a sequence of motivic invariants and studies their quantitative properties as well as their geometric interpretation. The work aims to extend the existing connections with Gromov-Witten theory, stable pairs theory, and modular forms for these new invariants, and to compare them with other motivic curve-counting invariants. One of the advantages of these new invariants is that they satisfy nice formulas and only depend on topological intersection numbers.