Spherical and Semibrick Classifications
Wahei Hara, Michael Wemyss
TL;DR
The paper surveys a unifying framework for classifying spherical objects, bricks, and simple-minded collections across diverse geometric settings, from Kleinian surface resolutions to 3-fold flops, via wall-crossing, mutations, and t-structure analysis. It develops a common strategy based on an invariant ell(x) and a wall-crossing apparatus (mutations along walls) to reduce objects to standard hearts, connecting bricks to simples through tau-tilting theory and contraction/algebraic hearts. Key contributions include precise classifications in finite geometric settings, the Bridgeland stability/BDL approach, the silting-discrete paradigm with the König–Yang bijection, and explicit results in symplectic and affine settings (Ishii–Uehara, Keating–Smith, Shimpi). The work highlights deep links between derived categories, noncommutative algebras (preprojective, contraction, and Jacobi algebras), and wall-crossing phenomena, with broad implications for autoequivalences and stability manifolds.
Abstract
This article provides an overview of the techniques related to classification of spherical and more general objects within triangulated categories, and its relationship with algebraic geometry, representation theory and symplectic geometry. The primary focus are the techniques of the authors within the 'finite' algebraic geometry setting in dimensions two and three, and within silting discrete algebras, but other approaches including to more general settings by Bapat-Deopurkar-Licata, Smith-Wemyss, Ishii-Uehara, Keating-Smith and Shimpi are also surveyed, in varying levels of detail. Various explicit examples are provided.