Gulbrandsen, Martin G.; Halle, Lars H.; Hulek, Klaus

A GIT Construction of Degenerations of Hilbert Schemes of Points

Doc. Math. 24, 421-472 (2019)
DOI: 10.25537/dm.2019v24.421-472


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.

Mathematics Subject Classification

14D06, 14C05, 14L24, 14D23


GIT, geometric invariant theory, degeneration, Hilbert scheme, expanded degenerations, Donaldson-Thomas invariants, Pandharipande-Thomas stable pairs, $K3$ surfaces, Hilbert scheme of points, Lie-Wu stack, bipartite graph, smooth support, numerical support, GIT quotient, stack quotient, Deligne-Mumford stack


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Gulbrandsen, Martin G.
University of Stavanger, Department of Mathematics and Physics, 4036 Stavanger, Norway
Halle, Lars H.
University of Copenhagen, Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark
Hulek, Klaus
Leibniz Universit\``at Hannover, Institut f\''ur Algebraische Geometrie, Welfengarten 1, 30060 Hannover, Germany