Stability of fibrations over one-dimensional bases, and standard models of del Pezzo fibrations
We introduce a notion of stability for varieties fibered over curves, motivated by Kollár's stability for homogeneous polynomials with integer coefficients. We analyze geometric implications of stability for 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.