Bricks and stones in the homological minimal model programme

Shimpi, Parth (2026) Bricks and stones in the homological minimal model programme. PhD thesis, University of Glasgow.

Abstract

Bricks and stones, morphisms between them and their cones, often build and control the symmetries of many categories arising in algebra and geometry. This thesis classifies such objects and related structures within categories arising from mild singularities in dimensions two and three.

More precisely, we classify t-structures on the local derived category of a 3-fold flopping contraction that are intermediate with respect to the heart of perverse coherent sheaves. Equivalently, this describes the complete lattice of torsion classes for the associated modification algebra. The intermediate hearts are all given as

(1) categories of coherent sheaves on birational models and tilts thereof in skyscrapers,
(2) algebraic t-structures described in the homological minimal model programme, or
(3) combinations of the above over appropriate open covers.

An analogous classification is also proved for (minimal as well as partial) resolutions of Kleinian singularities, thus providing a description of all torsion pairs in the module categories of (contracted) affine preprojective algebras.

The results have immediate applications to the classification of spherical modules and (semi)bricks, and are first steps towards describing all t-structures and spherical objects in the derived categories.

Such a programme is executed for flopping contractions with irreducible exceptional fibres. We show that any complex of coherent sheaves which admit no negative self-extensions is, up to flops and mutation equivalences, either

(1) a module over a derived–equivalent algebra, or
(2) a two–term extension of a coherent sheaf by skyscraper sheaves, or
(3) a direct sum of shifts of skyscrapers.

Consequently classifications of bricks, spherical objects, stability conditions, and algebraic t-structures in the local derived category are obtained; the lists populated by the homological minimal programme turn out exhaustive. We deduce that the Bridgeland stability manifold is connected, and that all basic tilting complexes on the variety (equivalently on the associated g−tame algebras) are related by shifts and iterated mutation.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Wemyss, Professor Michael
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86142
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 17 Jul 2026 11:07
• Last Modified: 17 Jul 2026 13:17
• URI: https://theses.gla.ac.uk/id/eprint/86142