Details
| Original language | English |
|---|---|
| Pages (from-to) | 1-56 |
| Number of pages | 56 |
| Journal | J ALGEBRAIC GEOM |
| Volume | 30 |
| Issue number | 1 |
| Early online date | 29 Jun 2020 |
| Publication status | Published - 2021 |
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.
Keywords
- math.AG
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
- Mathematics(all)
- Algebra and Number Theory
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In: J ALGEBRAIC GEOM, Vol. 30, No. 1, 2021, p. 1-56.
Research output: Contribution to journal › Article › Research › peer review
}
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
N1 - Funding Information: Received February 18, 2018, and, in revised form, June 25, 2019. 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.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - math.AG
UR - http://www.scopus.com/inward/record.url?scp=85103285734&partnerID=8YFLogxK
U2 - 10.1090/jag/765
DO - 10.1090/jag/765
M3 - Article
VL - 30
SP - 1
EP - 56
JO - J ALGEBRAIC GEOM
JF - J ALGEBRAIC GEOM
SN - 1056-3911
IS - 1
ER -