Noncommutative deformations and the birational geometry of 3-folds -- Will Donovan, November 8, 2013
I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. I will give a brief introduction to noncommutative deformations, and explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations. This construction yields new information about the (commutative) geometry of the 3-fold, and provides a new tool to differentiate between flops. As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve, relating a Fourier-Mukai flop-flop functor and a spherical twist about the universal family over the noncommutative deformation algebra. I will also explain work in progress applying this approach to other geometric situations, and to higher dimensions.