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、物理和工程的其他部分有许多应用。代数几何中一些研究得很好的对象是射影平面(具有附加的“无穷远点”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目，例如通过固定点或具有特殊形状，这是称为计数几何的代数几何分支的起点。在过去的二十年里，在物理学中，计数几何和弦理论的相互作用产生了许多令人振奋的新发展，并成为当前研究的一个中心领域。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，在一般代数曲面上，与数学和物理都有联系。这些不变量包含有关奇异曲线的数量、曲面的几何性质以及奇异曲线的参数化空间的复杂信息。该项目设想开发用于计算的理论框架和工具，以及在数学不同学科之间建立联系。它将加深对代数簇几何的理解，并将其应用于弦理论中的时空描述。本研究项目研究了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动机推广。以前的工作已经证明，具有某些给定奇点的变种数满足普适公式。这个项目涉及到在动机意义上对这些普遍公式的推广。本研究定义了一系列动机不变量，并研究了它们的数量性质以及它们的几何解释。这项工作旨在扩展现有的与Gromov-Witten理论、稳定对理论和这些新不变量的模形式的联系，并将它们与其他基元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良好的公式，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.