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 -