Comparison of fundamental group schemes of a projective variety and an ample hypersurface

Abstract

Abstract. Let X be a smooth projective variety defined over an algebraically closed field, and let L be an ample line bundle over X. We prove that for any smooth hypersurface D on X in the complete linear system |L ⊗d |, the inclusion map D ֒→ X induces an isomorphism of fundamental group schemes, provided d is sufficiently large and dim X ≥ 3. If dim X = 2, and d is sufficiently large, then the induced homomorphism of fundamental group schemes remains surjective. We give an example to show that the homomorphism of fundamental group schemes induced by the inclusion map of a reduced ample curve in a smooth projective surface is not surjective in general.

On the Grothendieck–Lefschetz theorem for a family of varieties

“…By [1] and [10] we know that there exists n 1 (depending only on X ′ 0 ) such that for every d ≥ n 1 and for every Y ′ 0 ∈ |H ⊗d k | the homomorphism…”

“…is surjective when dim(Z) ≥ 2 and an isomorphism when dim(Z) ≥ 3, thus in particular if char(k) = 0 the same result automatically holds for the fundamental group scheme (even when k is not algebraically closed and this can be seen using the fundamental short exact sequence). When char(k) = p > 0 then in [10] and [1] it has been proved, independently, that theorems of Grothendieck-Lefschetz type hold in the following formulation: let H be a very ample line bundle over Z then when dim(Z) ≥ 2 the natural homomorphism ϕ : π 1 (Y, y) → π 1 (Z, y) between fundamental group schemes induced by the inclusion map ϕ : Y ֒→ Z is faithfully flat whenever Y is in the complete linear system |H ⊗d | for any integer d ≥ d 0 where d 0 is an integer depending only on Z. If moreover dim(Z) ≥ 3, then ϕ is an isomorphism whenever Y is in the complete linear system |H ⊗d | for any integer d ≥ d 1 where d 1 is an integer depending only on Z.…”

Essentially finite vector bundles on varieties with trivial tangent bundle

“…There is a natural bijective correspondence between the p-Lie algebras over k and the local group-schemes over k of height one [8, p. 139]. Let (1) G be the local group-scheme of height one corresponding to the p-Lie algebra…”

Toward a geometric analogue of Dirichlet’s unit theorem

“…An arithmetic R-divisor on Spec(O K ) is a pair (D, ξ) consisting of an R-divisor D on Spec(O K ) and ξ ∈ Ξ K . We often denote the pair (D, ξ) by D. The arithmetic principal R-divisor (x) R of x ∈ K × ⊗ R is the arithmetic R-divisor given by (x) R := ∑ P ord P (x)[P], −2L R (x) , where P runs over the set of all maximal ideals of O K and ord P (x) := a 1 ord P (x 1 ) + · · · + a r ord P (x r ) for x = x a 1 1 · · · x a r r (x 1 , . .…”

“…[5]). It is however a very interesting problem to find a sufficient condition for the existence of an arithmetic small R-section, that is, an element x such that x = x a 1 1 · · · x a r r (x 1 , . .…”