A GIT Construction of Degenerations of Hilbert Schemes of Points

Abstract

We present a Geometric Invariant Theory (GIT) con-struction which allows us to construct good projective degenerationsof Hilbert schemes of points for simple degenerations. A comparisonwith the construction of Li and Wu shows that our GIT stack andthestack they construct are isomorphic, as are the associated coarse mod-uli schemes. Our construction is sufficiently explicit to obtain goodcontrol over the geometry of the singular fibres. We illustrate thisby giving a concrete description of degenerations of degreenHilbertschemes of a simple degeneration with two components.