Logarithmic enumerative geometry and moduli spaces

模空间和计数几何是代数几何的中心主题。模空间描述了给定类型的所有几何对象。计数不变量是模空间的自然特征数。它们产生于研究这些空间的拓扑结构，并且通常以Urve计数问题为模型：受几何约束的空间中的曲线的计数。该提案旨在提出一种成熟的对数计数几何理论。对数几何用组合数据丰富了代数几何的对象。它通过一种称为退化的过程连接到传统的代数几何。对数结构已经被研究了几十年，但直到最近才发展出了解其组合复杂性的方法。这是热带几何主题的一部分。在这个对数方向上提出的进展是由计数几何的基本对象之间的相互作用指导的：曲线的模空间、稳定映射空间和嵌入曲线的希尔伯特格式。这些相互作用产生了显著的洞察力，例如Gromov-Witten(GW)/Donaldson-Thomas(DT)对应，GW理论的上同调场理论结构，以及许多关于曲线模空间的漂亮计算。我们提出了一个新的对数GW/DT猜想，并利用奥比诺兹研究了对数GW理论的代数结构，以及曲线的模空间上的一个新的对数交理论。这些结果将导致传统的计数几何中的结果，例如证明了非常大的一类Calabi-Yau三重数的标准GW/DT对应。它还将在不同的主题之间建立联系，例如镜像对称和曲面上点的希尔伯特方案。最后，它将为利用热带几何研究曲线的模空间开辟一个全新的方向。

Moduli spaces and enumerative geometry are central themes in algebraic geometry. Moduli spaces describe all geometric objects of a given type. Enumerative invariants are natural characteristic numbers of moduli spaces. They arise from studying the topology of these spaces and are often modeled on urve counting problems: counts of curves in a space subject to geometric constraints. The proposal aims to bring forward a fully-fledged theory of logarithmic enumerative geometry. Logarithmic geometry enriches objects of algebraic geometry with combinatorial data. It connects to traditional algebraic geometry by a process called degeneration. Logarithmic structures have been studied for decades but only recently have methods been developed to understand their combinatorial complexity. This is part of the subject of tropical geometry.The proposed advances in this logarithmic direction are guided by the interactions between the basic objects of enumerative geometry: the moduli space of curves, the space of stable maps, and the Hilbert scheme of embedded curves. The interactions have led to remarkable insights, such as the Gromov-Witten (GW)/Donaldson-Thomas (DT) correspondence, the cohomological field theory structure of GW theory, and numerous beautiful calculations on the moduli space of curves. We propose a new logarithmic GW/DT conjecture, a study of the algebraic structure of logarithmic GW theory using orbifolds, and a new logarithmic intersection theory on the moduli space of curves. These will lead to results in traditional enumerative geometry, such as a proof of the standard GW/DT correspondence for a very large class of Calabi-Yau threefolds. It will also build connections between different themes, such as mirror symmetry and the Hilbert schemes of points on a surface. Finally, it will open an entirely new direction in the study of the moduli space of curves using tropical geometry.