Hilbert schemes of log points

代数簇的退化是模空间紧化、镜像对称性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一个表现良好的退化K3曲面族的第二类退化K3曲面点的Hilbert格式。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。拟议中的建筑灵感来自对数几何的最新进展。特别地，我们的目的是在一般的简单正规交叉对上构造一个对数点的Hilbert格式。最终目的是为K3曲面的III型退化点的Hilbert格式构造行为良好的退化点。这将产生超Kähler品种的III型退化的具体例子，并提供关于Hyperkähler品种如何适应Gross-Siebert Mirror对称性计划的洞察。这些例子及其单列表示也将在算术环境中研究。

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.