Title: Kollár stability of low degree del Pezzo fibrations
Abstract: I will introduce a notion of Kollár stability for families of projective varieties fibered over one-dimensional bases, a common generalization of Tate's minimal models for elliptic curves and Kollár's stability for hypersurfaces over PIDs. I will then explain how a correct choice of stability conditions leads to good models of low degree threefold del Pezzo fibrations over curves. Applications will include a proof of Corti's conjecture on the existence of standard models of degree 1 del Pezzo fibrations, and an analog of Tate's minimal models for degree 1 del Pezzo fibrations. This is joint work with Hamid Abban and Igor Krylov.