Details
| Original language | English |
|---|---|
| Pages (from-to) | 589-647 |
| Number of pages | 59 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 377 |
| Issue number | 1 |
| Early online date | 19 Oct 2023 |
| Publication status | Published - 2024 |
Abstract
We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.
Keywords
- Birational geometry, derived categories, geometric invariant theory, Hilbert schemes of points, stability conditions
ASJC Scopus subject areas
- Mathematics(all)
- Mathematics(all)
- Applied Mathematics
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In: Transactions of the American Mathematical Society, Vol. 377, No. 1, 2024, p. 589-647.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Variation of Stability for Moduli Spaces of Unordered Points in the Plane
AU - Gallardo, Patricio
AU - Schmidt, Benjamin
N1 - Funding Information: Received by the editors July 18, 2022, and, in revised form, March 15, 2023, and May 23, 2023. 2020 Mathematics Subject Classification. Primary 14C05; Secondary 14E30, 14F08, 14L24. Key words and phrases. Birational geometry, geometric invariant theory, derived categories, Hilbert schemes of points, stability conditions. The second author was supported by an AMS-Simons travel grant during part of this work. The first author was supported by the University of California, Riverside and Washington University at St Louis.
PY - 2024
Y1 - 2024
N2 - We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.
AB - We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.
KW - Birational geometry
KW - derived categories
KW - geometric invariant theory
KW - Hilbert schemes of points
KW - stability conditions
UR - http://www.scopus.com/inward/record.url?scp=85182584779&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2205.15238
DO - 10.48550/arXiv.2205.15238
M3 - Article
AN - SCOPUS:85182584779
VL - 377
SP - 589
EP - 647
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 1
ER -