On the Grothendieck-Lefschetz Theorem for a Family of Varieties

TL;DR

This paper extends the Grothendieck-Lefschetz theorem to families of varieties over fields of positive characteristic, establishing conditions under which fundamental group schemes of hypersurfaces relate faithfully or isomorphically to those of the ambient variety.

Contribution

It proves new conditions for the faithful flatness and isomorphism of fundamental group schemes in families of varieties over Witt rings in positive characteristic.

Findings

Existence of a threshold degree d_0 for faithful flatness of fundamental group schemes.

Existence of a threshold degree d_1 for isomorphism of fundamental group schemes.

Results depend only on the geometry of the original variety, not on specific hypersurfaces.

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.