Stability of fibrations over one-dimensional bases

Abstract

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 one and two del Pezzo fibrations, settling a conjecture of Corti from 1996.

34 pages, v2: minor typos corrected. A reference to Loginov's paper arXiv:1710.02482 is added. v3: note the first author's name change, a major revision, with the proof of terminality rewritten and now valid in most positive characteristics, accepted at Duke Math J