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.
DOI: https://doi.org/10.1090/jag/765
Published electronically: June 29, 2020
Full-text PDF

Abstract | References | Additional Information

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.


References [Enhancements On Off] (What's this?)


Additional Information

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

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

Klaus Hulek
Affiliation: Leibniz Universität Hannover, Institut für algebraische Geometrie, Welfengarten 1, 30167 Hannover, Germany
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
Email: zhangziyu@shanghaitech.edu.cn

DOI: https://doi.org/10.1090/jag/765
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.