Stability of fibrations over one-dimensional bases

TL;DR

This paper introduces a new stability concept for fibered varieties over curves, develops tools to analyze their birational models, and proves the existence of standard models for certain threefold fibrations, confirming a longstanding conjecture.

Contribution

It proposes a novel stability notion for fibrations over curves and establishes the existence of standard models for specific threefold del Pezzo fibrations.

Findings

Established a new stability framework for fibered varieties.

Proved the existence of standard models for degree one and two del Pezzo fibrations.

Settled Corti's 1996 conjecture on threefold fibrations.

Abstract

We introduce and study a new notion of stability for varieties fibered over curves, motivated by Koll\'ar'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.