Gulbrandsen-Halle-Hulek degeneration and Hilbert-Chow morphism

TL;DR

This paper proves that two different models of degenerations of Hilbert schemes of points on surfaces, previously constructed separately, are actually isomorphic in the case of semistable degenerations without triple points.

Contribution

The paper establishes the isomorphism between Gulbrandsen-Halle-Hulek's and the author's models of Hilbert scheme degenerations for certain surface degenerations.

Findings

Models are isomorphic in the semistable case without triple points.

Unifies two previously distinct approaches to Hilbert scheme degenerations.

Provides a clearer understanding of degeneration behavior in algebraic geometry.

Abstract

For a semistable degeneration of surfaces without a triple point, we show that two models of degeneration of Hilbert scheme of points of the family, Gulbrandsen-Halle-Hulek degeneration given in [GHH] and the one given by the author in [N], are actually isomorphic.