On homotopy exact sequences for normal schemes

Let \(f: X \rightarrow S\) $f:X\to S$ be a surjective morphism of finite type between connected locally Noetherian normal schemes. We discuss sufficient conditions that the sequence of the étale fundamental groups \(\begin{aligned} \pi _{1}(X\times _{S}{\overline{\eta }},*) \rightarrow \pi _{1}(X,*) \rightarrow \pi _{1}(S,*)\rightarrow 1 \end{aligned}\) $$\begin{array}{c}\hfill {\pi }_1(X{\times }_S\overline{\eta },\ast )\to {\pi }_1(X,\ast )\to {\pi }_1(S,\ast )\to 1\end{array}$$ is exact, where \({\overline{\eta }}\) $\overline{\eta }$ is a geometric generic point of S and \(*\) $\ast$ is a geometric point of \(X\times _{S}{\overline{\eta }}\) $X{\times }_S\overline{\eta }$ . In the present paper, we generalize those in (Grothendieck and Raynaud in Séminaire de Géometrie Algébrique du Bois Marie 1960/61, Revétements Etales et Groupe Fondamental (SGA 1), Lecture Notes in Mathematics, vol. 224. Springer, Berlin, 1971; Hoshi in J Math Sci Univ Tokyo 21(2):153–219, 2014), and (Mitsui in Algebra Number Theory 9(5):1089–1136, 2015). We show that the conditions we give are also necessary conditions in the case where, for instance, S is an affine smooth curve over a field of characteristic 0.