# A GIT Construction of Degenerations of Hilbert Schemes of Points

### Martin G. Gulbrandsen

University of Stavanger, Department of Mathematics and Physics, 4036 Stavanger, Norway### Lars H. Halle

University of Copenhagen, Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark### Klaus Hulek

Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten 1, 30060 Hannover, Germany

## Abstract

We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT stack and the stack they construct are isomorphic, as are the associated coarse moduli schemes. Our construction is sufficiently explicit to obtain good control over the geometry of the singular fibres. We illustrate this by giving a concrete description of degenerations of degree $n$ Hilbert schemes of a simple degeneration with two components.

## Cite this article

Martin G. Gulbrandsen, Lars H. Halle, Klaus Hulek, A GIT Construction of Degenerations of Hilbert Schemes of Points. Doc. Math. 24 (2019), pp. 421–472

DOI 10.4171/DM/685