On homotopy exact sequences for normal schemes

TL;DR

This paper investigates the conditions under which the homotopy exact sequence of étale fundamental groups is exact for morphisms between normal schemes, providing new characterizations especially for smooth varieties over curves.

Contribution

It offers new criteria for the exactness of the homotopy sequence and characterizes the kernel's topological finite generation in specific morphisms.

Findings

Identifies conditions for exactness of the étale fundamental group sequence.

Provides a characterization of the kernel's topological finite generation.

Applies results to morphisms from smooth varieties to smooth curves.

Abstract

Consider a morphism between connected locally Noetherian normal schemes. In this paper, we discuss when the sequence of the etale fundamental groups associated to the morphism is exact. Moreover, we give a characterization of when the kernel of the induced homomorphism between their fundamental groups is topologically finitely generated, for the morphism from a smooth variety to a smooth curve.