A GIT construction of degenerations of Hilbert schemes of points

TL;DR

This paper introduces a GIT-based method to construct and analyze degenerations of Hilbert schemes of points, providing explicit control over singular fibers and establishing isomorphisms with existing constructions.

Contribution

The authors develop a GIT construction for degenerations of Hilbert schemes of points that aligns with Li and Wu's approach and offers detailed geometric insights.

Findings

Constructed projective degenerations of Hilbert schemes using GIT.

Established isomorphism between the GIT stack and Li-Wu's stack.

Provided explicit descriptions of degenerations in simple cases.

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.