Let f:X -> S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: X \rightarrow S$$\end{document} be a surjective morphism of finite type between connected locally Noetherian normal schemes. We discuss sufficient conditions that the sequence of the & eacute;tale fundamental groups pi 1(XxS eta<overline>,& lowast;)->pi 1(X,& lowast;)->pi 1(S,& lowast;)-> 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi _{1}(X\times _{S}{\overline{\eta }},*) \rightarrow \pi _{1}(X,*) \rightarrow \pi _{1}(S,*)\rightarrow 1 \end{aligned}$$\end{document}is exact, where eta<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\eta }}$$\end{document} is a geometric generic point of S and & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} is a geometric point of XxS eta<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\times _{S}{\overline{\eta }}$$\end{document}. In the present paper, we generalize those in (Grothendieck and Raynaud in S & eacute;minaire de G & eacute;ometrie Alg & eacute;brique du Bois Marie 1960/61, Rev & eacute;tements 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.
