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.
I: Journal of Algebraic Geometry, Bind 30, Nr. 1, 2021, s. 1 - 56.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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