April 2019, Boston

**Title:** Parameter spaces of del Pezzo fibrations and birational geometry

**Abstract:**

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.