May 2019, Providence
Title: Parameter spaces of del Pezzo surfaces and birational geometry of del Pezzo fibrations
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 together with a notion of stability. Then I define what are the best models of del Pezzo fibrations of degrees 1 and 2 and show the existence of a good birational model.