April 2018, Bayreuth

**Title:**Stability over rings and good models of del Pezzo fibrations.

**Abstract:**This talk is motivated by the following problem, given a three-dimensional Mori fiber space, can we find a birational to it model with nice singularities? Sarkisov proved that for a conic bundle there exists a smooth model. For del Pezzo fibrations smooth model may not exist in case of degree <4. Corti has shown that there are Gorenstein (resp. 2-Gorenstein) models for del Pezzo fibrations of degree 3 (resp. 2). He proved it by constructing explicit birational maps improving singularities. Kollar improved his result in degree 3 using geometric invariant theory. I discuss what are the issues in adapting Kollar's approach for degrees 1 and 2 and how to work around them.