Expansions for Hilbert schemes of points on semistable degenerations

Abstract: Let $X\rightarrow C$ be a projective family of surfaces over a curve with smooth general fibres and simple normal crossing singularity in the special fibre $X_0$. We construct a good compactification of the moduli space of relative length $n$ zero-dimensional subschemes on $X\setminus X_0$ over $C\setminus\{0\}$. In order to produce this compactification we study expansions of the special fibre $X_0$ together with various GIT stability conditions, generalising the work of Gulbrandsen-Halle-Hulek who use GIT to offer an alternative approach to the work of Li-Wu for Hilbert schemes of points on simple degenerations. We construct stacks which we prove to be equivalent to the underlying stack of some choices of logarithmic Hilbert schemes produced by Maulik-Ranganathan.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

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