The geometry of degenerations of Hilbert schemes of points

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The geometry of degenerations of Hilbert schemes of points. / Gulbrandsen, Martin G.; Halle, Lars H.; Hulek, Klaus; Zhang, Ziyu.

In: Journal of Algebraic Geometry, Vol. 30, No. 1, 2021, p. 1 - 56.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Gulbrandsen, MG, Halle, LH, Hulek, K & Zhang, Z 2021, 'The geometry of degenerations of Hilbert schemes of points', Journal of Algebraic Geometry, vol. 30, no. 1, pp. 1 - 56. https://doi.org/10.1090/jag/765

APA

Gulbrandsen, M. G., Halle, L. H., Hulek, K., & Zhang, Z. (2021). The geometry of degenerations of Hilbert schemes of points. Journal of Algebraic Geometry, 30(1), 1 - 56. https://doi.org/10.1090/jag/765

Vancouver

Gulbrandsen MG, Halle LH, Hulek K, Zhang Z. The geometry of degenerations of Hilbert schemes of points. Journal of Algebraic Geometry. 2021;30(1):1 - 56. https://doi.org/10.1090/jag/765

Author

Gulbrandsen, Martin G. ; Halle, Lars H. ; Hulek, Klaus ; Zhang, Ziyu. / The geometry of degenerations of Hilbert schemes of points. In: Journal of Algebraic Geometry. 2021 ; Vol. 30, No. 1. pp. 1 - 56.

Bibtex

@article{03bf9d6ee6d64f149a0ea0c64172123c,
title = "The geometry of degenerations of Hilbert schemes of points",
abstract = "Given a strict simple degeneration f : X → C the first three authors previously constructed a degeneration IX/Cn → 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 IX/Cn → 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 (IX/Cn )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 IX/Cn → C carries a nowhere degenerate relative logarithmic 2-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.",
author = "Gulbrandsen, {Martin G.} and Halle, {Lars H.} and Klaus Hulek and Ziyu Zhang",
year = "2021",
doi = "10.1090/jag/765",
language = "English",
volume = "30",
pages = "1 -- 56",
journal = "Journal of Algebraic Geometry",
issn = "1056-3911",
publisher = "American Mathematical Society",
number = "1",

}

RIS

TY - JOUR

T1 - The geometry of degenerations of Hilbert schemes of points

AU - Gulbrandsen, Martin G.

AU - Halle, Lars H.

AU - Hulek, Klaus

AU - Zhang, Ziyu

PY - 2021

Y1 - 2021

N2 - Given a strict simple degeneration f : X → C the first three authors previously constructed a degeneration IX/Cn → 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 IX/Cn → 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 (IX/Cn )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 IX/Cn → C carries a nowhere degenerate relative logarithmic 2-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.

AB - Given a strict simple degeneration f : X → C the first three authors previously constructed a degeneration IX/Cn → 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 IX/Cn → 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 (IX/Cn )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 IX/Cn → C carries a nowhere degenerate relative logarithmic 2-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.

U2 - 10.1090/jag/765

DO - 10.1090/jag/765

M3 - Journal article

VL - 30

SP - 1

EP - 56

JO - Journal of Algebraic Geometry

JF - Journal of Algebraic Geometry

SN - 1056-3911

IS - 1

ER -

ID: 244327436