A note on stability conditions on product-quotient surfaces
Abstract
Let X be a product-quotient surface, which is by definition a minimal resolution of (C1× C2)/G, where Ci are smooth curves of genus ≥ 2 and G is a finite group. When G acts freely, we show that every product Bridgeland stability condition on C1× C2 induces a tilt stability condition on X, and that, if pg(X)=q(X)=0, every tilt stability condition on X arises this way. When (C1× C2)/G has only A1-singularities, we use induction \cite{MMS} and derived McKay correspondence \cite{BKR} to produce non-geometric stability conditions σX on X, which satisfy the full support property and lie in the closure of the geometric chamber. Whether σX also lies in the closure of the GL̃+(2,ℝ)-orbit of the tilt stability conditions depends on a single inequality, ωX2 ≥ n/8, where n is the number of A1-singularities. When it fails, the geometric chamber is strictly larger than this orbit.
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BibTeX
@misc{dutta2026note,
title = {{A note on stability conditions on product-quotient surfaces}},
author = {Uttaran Dutta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15276},
eprint = {2610.15276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15276},
}
RIS
TY - UNPB
TI - A note on stability conditions on product-quotient surfaces
AU - Uttaran Dutta
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15276
UR - https://arxiv.org/abs/2610.15276
AB - Let X be a product-quotient surface, which is by definition a minimal resolution of (C_1× C_2)/G, where C_i are smooth curves of genus ≥ 2 and G is a finite group. When G acts freely, we show that every product Bridgeland stability condition on C_1× C_2 induces a tilt stability condition on X, and that, if p_g(X)=q(X)=0, every tilt stability condition on X arises this way. When (C_1× C_2)/G has only A_1-singularities, we use induction \cite{MMS} and derived McKay correspondence \cite{BKR} to produce non-geometric stability conditions σ_X on X, which satisfy the full support property and lie in the closure of the geometric chamber. Whether σ_X also lies in the closure of the GL̃^+(2,ℝ)-orbit of the tilt stability conditions depends on a single inequality, ω_X^2 ≥ n/8, where n is the number of A_1-singularities. When it fails, the geometric chamber is strictly larger than this orbit.
ER -