Abstract

I will explain why, when studying derived autoequivalences of 3-folds, it is necessary to consider noncommutative deformations of curves. In the talk I will give a construction of a certain “noncommutative twist” associated to any floppable curve that recovers the flop-flop functor on the level of the derived category. The idea is that the commutative deformation base is too small for the homological algebra to work, so we need to fatten it by considering noncommutative directions. This generalizes work of Seidel—Thomas and Toda who considered the special case when the curve deforms in only one direction.

I will try to explain why considering noncommutative deformations is strictly necessary, as I will show that considering only the commutative deformations does not give a derived autoequivalence as one might hope. The talk will be based around one example, where the birational geometry of a certain 3-fold is controlled by the cusp in the quantum plane, which is a 9-dimensional self-injective algebra. This is all based on joint work with Will Donovan.