On the Grothendieck–Lefschetz theorem for a family of varieties

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 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 d0, depending only on X, such that for any d⩾d0, any Y∈|H⊗d| connected and smooth over W and any y∈Y(W) the natural R-morphism of fundamental group schemes π1(YR,yR)→π1(XR,yR) is faithfully flat, XR, YR, yR being respectively the pull back of X, Y, y over Spec(R). If moreover dim(X/W)⩾3 then there exists an integer d1, depending only on X, such that for any d⩾d1, any Y∈|H⊗d| connected and smooth over W and any section y∈Y(W) the morphism π1(YR,yR)→π1(XR,yR) is an isomorphism.