Hilbert schemes of log points

对数点的希尔伯特方案

• 批准号: 516701553
• 负责人: Professor Dr. Christian Liedtke
• 金额: --
• 依托单位: Lehr- und Forschungseinheit Algebra (M11)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/516701553?language=en
• 关键词: Hilbert; schemes; log; points; Hilbert; schemes; log; points

Degenerations of algebraic varieties are key ingredients in the compactification of moduli spaces, in mirror symmetry, and the computation of enumerative invariants. Gulbrandsen, Halle and Hulek constructed a well-behaved degenerating family of Hilbert schemes of points of a Type II degenerating family of K3 surfaces using expanded degenerations. The goal of this project is to give an alternative construction of this degeneration using log-theoretic methods. The proposed construction is inspired by recent advances in logarithmic geometry. In particular, we aim to construct a Hilbert scheme of logarithmic points on a general simple normal crossing pair. The ultimate goal is to construct well-behaved degenerations also for Hilbert schemes of points of Type III degenerations of K3 surfaces. This will yield concrete examples of Type III degenerations of hyperkähler varieties and provide insight into how hyperkähler varieties fit into the Gross-Siebert Mirror Symmetry program. These examples and their monodromy representations will also be studied in the arithmetic context.

代数变化的退化是模量空间，镜像对称性和计算枚举不变性的关键成分。古尔布兰森（Gulbrandsen），哈雷（Halle）和赫勒克（Hulek）使用扩展的退化构建了一个表现得很好的希尔伯特（Hilbert）方案的堕落家族。该项目的目的是使用对数理论方法对这种变性进行替代结构。所提出的结构的灵感来自对数几何形状的最新进展。特别是，我们旨在在一般简单的正常交叉对上构建对数点的希尔伯特方案。最终的目标是为K3表面III型退化点的Hilbert方案构建行为良好的退化。这将产生Hyperkähler品种的III型变性的具体实例，并提供有关Hyperkähler品种如何适合Gross-Siebert Mirror对称程序的洞察力。这些示例及其单肌表示也将在算术背景下进行研究。