Stability of fibrations over one-dimensional bases

We introduce and study a new notion of stability for varieties fibered over curves, motivated by Kollár’s stability for homogeneous polynomials with integral coefficients. We develop tools to study 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 1 del Pezzo fibrations, settling a conjecture of Corti.