Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The geometry of degenerations of Hilbert schemes of points


Authors: Martin G. Gulbrandsen, Lars H. Halle, Klaus Hulek and Ziyu Zhang
Journal: J. Algebraic Geom. 30 (2021), 1-56
DOI: https://doi.org/10.1090/jag/765
Published electronically: June 29, 2020
MathSciNet review: 4233177
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Abstract: Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ 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 $I^n_{X/C} \to C$ is a dlt model. This is even a good minimal dlt model if $f \colon X \to C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of K3 surfaces we show that the stack ${\mathcal I}^n_{X/C} \to C$ carries a nowhere degenerate relative logarithmic $2$-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.


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Martin G. Gulbrandsen
Affiliation: Department of Mathematics and Natural Sciences, University of Stavanger, 4036 Stavanger, Norway
MR Author ID: 741161
Email: martin.gulbrandsen@uis.no

Lars H. Halle
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
MR Author ID: 899512
Email: larshhal@math.ku.dk

Klaus Hulek
Affiliation: Leibniz Universität Hannover, Institut für algebraische Geometrie, Welfengarten 1, 30167 Hannover, Germany
MR Author ID: 89705
Email: hulek@math.uni-hannover.de

Ziyu Zhang
Affiliation: Leibniz Universität Hannover, Institut für algebraische Geometrie, Welfengarten 1, 30167 Hannover, Germany
Address at time of publication: ShanghaiTech University, Institute of Mathematical Sciences, 393 Middle Huaxia Road, Shanghai 201210, People’s Republic of China
MR Author ID: 1021717
Email: zhangziyu@shanghaitech.edu.cn

Received by editor(s): February 18, 2018
Received by editor(s) in revised form: June 25, 2019
Published electronically: June 29, 2020
Additional Notes: The first author thanks the Research Council of Norway for partial support under grant 230986. The third author is grateful to DFG for partial support under grant Hu 337/7-1.
Article copyright: © Copyright 2020 University Press, Inc.