Aditya Khurmi: Smoothing of a rational singularity with non-finitely generated canonical algebra.

Let H  be an isolated surface singularity with a one-parameter smoothing X. If the canonical algebra R of X is finitely generated (as an O_X-algebra) it can be used to define a small modification Y = Proj R of X such that K_Y is Q-Cartier. This is used in the construction of flips in the minimal model program. 

As discovered by Kollár--Shepherd-Barron in 1988, the finite generation holds when H is a quotient singularity. The strict transform of H is a so-called P-resolution, and all such can be determined combinatorially. They proved that Q-Gorenstein deformations of these P-resolutions describe the components of the versal deformation space of H.

János Kollár conjectured that an analogous result holds when H is a rational singularity. In this talk, I will describe an approach, inspired by a paper of Cutkosky, to disprove the conjecture by constructing a smoothing with non-finitely generated canonical algebra, assuming a hypothesis we expect to be true.