A criterion for extending morphisms from open subsets of smooth fibrations of algebraic varieties

TL;DR

This paper establishes a new criterion for extending morphisms from open subsets to entire smooth fibrations of algebraic varieties, generalizing Zariski's main theorem to broader contexts.

Contribution

It introduces a sufficient condition for extending morphisms from open subsets of smooth fibrations to the whole variety, expanding the applicability of extension theorems in algebraic geometry.

Findings

Provides a criterion analogous to Zariski's main theorem for morphism extension.

Applicable to smooth morphisms and proper morphisms of algebraic varieties.

Enhances understanding of morphism extension in algebraic geometry.

Abstract

Given a smooth morphism $Y\to S$ and a proper morphism $P\to S$ of algebraic varieties we give a sufficient condition for extending an $S$-morphism $U\to P$, where $U$ is an open subset of $Y$, to an $S$-morphism $Y\to P$, analogous to Zariski's main theorem.