Noncommutative deformations and flops

The thesis will focus on the paper by Donovan and Wemyss, Noncommutative deformations and flops. In this work, the authors prove that the functor of noncommutative deformations is representable for every flipping or flopping irreducible rational curve in a 3-fold. As a result, each such curve can be associated with a noncommutative deformation algebra, denoted \(A_{\mathsf{con}}\). This new invariant both extends and unifies existing invariants for flopping curves in 3-folds, including Reid’s width and the bidegree of the normal bundle, and is applicable in contexts involving flips and singular schemes.

Prerequisites: Familiarity with scheme theory and derived categories.

Reference: Donovan-Wemyss, Noncommutative deformations and flops.