Variation of Stability for Moduli Spaces of Unordered Points in the Plane

authored by: Patricio Gallardo, Benjamin Schmidt
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.
Organisation(s): Institute of Algebraic Geometry
External Organisation(s): University of California at Riverside
Type: Article
Journal: Transactions of the American Mathematical Society
Volume: 377
Pages: 589-647
No. of pages: 59
ISSN: 0002-9947
Publication date: 2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.48550/arXiv.2205.15238 (Access: Open)
https://doi.org/10.1090/tran/9030 (Access: Closed)

BibTeX

@article{babf9dc8a0804c3bb8d52d52ad7ba8bc,
title = "Variation of Stability for Moduli Spaces of Unordered Points in the Plane",
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",
author = "Patricio Gallardo and Benjamin Schmidt",
note = "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. ",
year = "2024",
doi = "10.48550/arXiv.2205.15238",
language = "English",
volume = "377",
pages = "589--647",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "1",
}