Hilbert schemes of log points

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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.