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 : = 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 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.