Stability of fibrations over one-dimensional bases, and standard models of del Pezzo fibrations

We introduce and study a notion of stability for varieties fibered over curves, motivated by Koll\'ar's stability for homogeneous polynomials with integer coefficients. We develop tools that in principle give a complete understanding of geometric properties of stable birational models of fibrations whose fibers are complete intersections in weighted projective spaces. As an application, we prove the existence of standard models of threefold degree one and two del Pezzo fibrations, settling a conjecture of Corti from 1996. This is joint work with Hamid Ahmadinezhad and Igor Krylov.