The geometry of degenerations of Hilbert schemes of points

authored by

Martin G. Gulbrandsen, Lars H. Halle, Klaus Hulek, Ziyu Zhang

Abstract

Given a strict simple degeneration f : X → C the first three authors previously constructed a degeneration I

_X/C

ⁿ → C of the relative degree n Hilbert scheme of 0-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of f is at most 2. In this case we show that I

_X/C

ⁿ → C is a dlt model. This is even a good minimal dlt model if f : X → C has this property. We compute the dual complex of the central fibre (I

_X/C

ⁿ )

₀ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of K3 surfaces we show that the stack I

_X/C

ⁿ → C carries a nowhere degenerate relative logarithmic 2-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.

Organisation(s)

Institut für Algebraische Geometrie
Riemann Center for Geometry and Physics

External Organisation(s)

University of Stavanger
University of Copenhagen

Type

Artikel

Journal

J ALGEBRAIC GEOM

Volume

30

Pages

1-56

No. of pages

56

ISSN

1056-3911

Publication date

2021

Publication status

Veröffentlicht

Peer reviewed

Yes

ASJC Scopus Sachgebiete

Geometrie und Topologie, Algebra und Zahlentheorie

Electronic version(s)

https://arxiv.org/abs/1802.00622 (Access: Offen)
https://doi.org/10.1090/jag/765 (Access: Geschlossen)