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

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.

代数簇的退化是模空间紧化、镜像对称和计数不变量计算的关键因素。Gulbrandsen，Halle和Hulek利用扩展退化构造了K3曲面的II型退化点族的退化Hilbert格式族。这个项目的目标是使用对数理论方法给出这种退化的另一种构造。建议的建设是灵感来自对数几何的最新进展。特别是，我们的目标是构建一个希尔伯特计划的对数点上的一般简单的正常交叉对。最终的目标是构造良好的退化也为希尔伯特计划的点的III型退化的K3曲面。这将产生hyperkähler品种的III型退化的具体例子，并提供洞察hyperkähler品种如何适应Gross-Siebert镜像对称程序。这些例子和它们的单值表示也将在算术上下文中进行研究。