2019/03/28 by Matt Booth, Booth, Matt
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1903.12156
openalex publication_date 2019/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using Braun-Chuang-Lazarev's derived quotient, we enhance the contraction algebra of Donovan-Wemyss to an invariant valued in differential graded algebras. Given an isolated contraction X → Xcon of an irreducible rational curve C to a point p, we show that its derived contraction algebra controls the derived noncommutative deformations of C. We use dg singularity categories to prove that, when X is smooth, the derived contraction algebra recovers the geometry of Xcon complete locally around p, establishing a positive answer to a derived version of a conjecture of Donovan and Wemyss. When X → Xcon is a simple threefold flopping contraction, it is known that the Bridgeland-Chen flop-flop autoequivalence of Db(X) is a `noncommutative twist' around the contraction algebra. We show that the derived contraction algebra controls an analogous autoequivalence in more general settings, and in particular for partial resolutions of Kleinian singularities.