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 $dim(X/W)\ge 2$ there exists an integer $d_0$, depending only on $X$, such that for any $d\ge d_0$, 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 $dim(X/W)\ge 3$ then there exists an integer $d_1$, depending only on $X$, such that for any $d\ge d_1$, 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.

arXiv comment11 pages