Algebraic Geometry near-Boston Conference

April 2019, Boston

Title: Parameter spaces of del Pezzo fibrations and birational geometry

Del Pezzo fibrations are one of the types of the Mori Fiber Space output of the MMP. There may be many models for the del Pezzo fibration and we would like to work with the best one. For example it is known that for conic bundles there exists a model with a smooth total space. I will describe a construction of parameter space of del Pezzo surfaces of degree 1 and 2. Using this parameter space I define what are the best models of del Pezzo fibrations of degrees 1 and 2. Then I show the existence of a good birational model.

Iskovskih seminar in Steklov Institute

November 2019, Moscow

Title: The variety of semistable del Pezzo surfaces

Kollar introduced a generalization of GIT stability for hypersurfaces: stability over rings. This notion is useful for finding good reductions to finite characterstic and for finding good (semistable) birational models of fibrations. I will talk about extending this notion to del Pezzo surfaces of degree 1 and 2. These surfaces are hypersurfaces in weighted projective spaces, where GIT techniques do not work. On the other hand embeddings into the projective spaces are not complete intersections and are difficult to work with. I will talk about generalizations of stability in both of these settings. In particular, I will talk about parameter spaces of del Pezzo surfaces of degrees 1 and 2