Stability condition on a singular surface and its resolution

2024/11/29 by Chou, Tzu-Yang · 3 citations
Computer Science · Mathematics · #14F08 (Primary) 14J17 #18E40 #18G80 (Secondary) #Advanced Mathematical Modeling in Engineering #Algebraic Geometry (math.AG) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.19768

openalex publication_date 2024/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Let X be a surface with an ADE-singularity and let \widetildeX be its crepant resolution. In this paper, we show that there exists a Bridgeland stability condition σ_X on \rm D^b(X) and a weak stability condition σ_\widetildeX on the derived category of the desingularisation \rm D^b(\widetildeX), such that pushforward of σ_\widetildeX-semistable objects are σ_X-semistable We first construct Bridgeland stability conditions on \rm D^b(\widetildeX) associated to the contraction \widetildeX \longrightarrow X, generalizing the results of Tramel and Xia in \citeTX22, Then we deform it to a weak stability condition σ_\widetildeX and show that it descends to \rm D^b(X), producing the stability condition σ_X. Finally, we study the moduli spaces of σ_{π^∗ H,β,z}, of σ_\widetildeX, and of σ_X-semistable objects, and we show that the moduli spaces satisfy boundedness and openness, and hence are all Artin stacks of finite type over ℂ.

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• Stability conditions on a singular quadric threefold
• Autoequivalences and stability conditions on a degenerate K3 surface
• Some examples of moduli spaces on surfaces via wall-crossing

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