On the Grothendieck-Lefschetz theorem for a family of varieties

Abstract

Let k be an algebraically closed field of characteristic p > 0, W the ring of Witt vectors over k and R the integral closure of W in the algebraic closure (K) over bar of K := Frac(W); let moreover X be a smooth, connected and projective scheme over W and H a relatively very ample line bundle over X. We prove that when dim(X/W) >= 2 there exists an integer d(0), depending only on X, such that for any d >= do, any Y is an element of |H-circle times d| connected and smooth over W and any y is an element of Y(W) the natural R-morphism of fundamental group schemes pi(1)(Y-R. y(R)) -> pi(1)(X-R, y(R)) is faithfully flat, X-R. Y-R. y(R) being respectively the pull back of X. Y. y over Spec(R). If moreover dim(X/W) >= 3 then there exists an integer d(1), depending only on X. such that for any d >= d(1), any Y is an element of |H-circle times d| connected and smooth over W and any section y is an element of Y(W) the morphism pi(1)(Y-R. y(R)) -> pi(1)(X-R. y(R)) is an isomorphism. (C) 2011 Elsevier Masson SAS. All rights reserved.