Extending rationally connected fibrations from ample subvarieties

TL;DR

This paper proves a conjecture about extending rationally connected fibrations from ample subvarieties using deformation theory, with applications to Fano fibrations and classification of certain fiber structures.

Contribution

It establishes the extendability of morphisms from ample subvarieties with rationally connected fibers, advancing the understanding of fibrations in algebraic geometry.

Findings

Proves a conjecture of Sommese on morphism extendability.

Classifies projective bundle and quadric fibration structures on ample subvarieties.

Applies deformation theory to rational curves in the context of fibrations.

Abstract

Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.