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 ${\bar{K}}$ 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 $dim(X/{W})\geq 2$ there exists an integer $d_0$, depending only on $X$, such that for any $d\geq d_0$, any $Y\in |H^{\otimes d}|$ connected and smooth over ${W}$ and any $y\in Y({W})$ the natural ${R}$-morphism of fundamental group schemes $\pi_1(Y_R,y_R)\to \pi_1(X_R,y_R)$ is faithfully flat, $X_R$, $Y_R$, $y_R$ being respectively the pull back of $X$, $Y$, $y$ over $Spec(R)$. If moreover $dim(X/{W})\geq 3$ then there exists an integer $d_1$, depending only on $X$, such that for any $d\geq d_1$, any $Y\in |H^{\otimes d}|$ connected and smooth over ${W}$ and any section $y\in Y({W})$ the morphism $\pi_1(Y_R,y_R)\to \pi_1(X_R,y_R)$ is an isomorphism.