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 of K := F rac(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 ≥ d 0 , any Y ∈ |H ⊗d | connected and smooth over W and any y ∈ Y (W ) the natural R-morphism of fundamental group schemes π 1 (Y R , y R ) → π 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 ∈ |H ⊗d | connected and smooth over W and any section y ∈ Y (W ) the morphism π 1 (Y R , y R ) → π 1 (X R , y R ) is an isomorphism.Mathematics Subject Classification: 14J60, 14L15.