On the abelian fundamental group scheme of a family of varieties

Abstract

Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π 1 (X, x) ab → π 1 (Alb X/S , 0 Alb X/S ) induced by the canonical morphism from X to its Albanese scheme Alb X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS τ X/S ) ∨ → π 1 (X, x) ab → π 1 (Alb X/S , 0 Alb X/S ) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber Xη of X can be extended to a finite and commutative pointed torsor over X.

The action of the etale fundamental group scheme on the connected component of the essentially finite one

“…Let ∶ ←→ be the Abel-Jacobi (or Albanese) morphism sending to the origin . Then, we arrive at a commutative diagram in which the arrow is an isomorphism [1,Corollary 3.8]. Hence, as explained in Section 2.2,…”

“…Let be the Abel–Jacobi (or Albanese) morphism sending c to the origin e . Then, we arrive at a commutative diagram in which the arrow α is an isomorphism [, Corollary 3.8]. Hence, as explained in Section , Now let be the full subcategory of the category of affine group schemes defined by those which are commutative, finite and annihilated by .…”

“…The relation betweeńe t ( , ) and its celebrated predecessor, the etale fundamental group of [16] is quite simple: Let be an algebraic closure of , and write = ⊗ . Then, using the obvious geometric point ∶ Spec → , we construct the geometric fundamental group 1 ( , ) of .…”

Models of Torsors Over Affine Spaces

“…So far the problem of extending the G-torsor Y → X η has consisted in finding a finite and flat S-group scheme G ′ whose generic fibre is isomorphic to G and a G ′ -torsor T → X whose generic fibre is isomorphic to Y → X η as a G-torsor. Some solutions, from Grothendieck's first ideas until nowadays, are known in some particular relevant cases and are the object of many classical and well known results and more recent papers, see for instance [11,Exposé X], [17, §3], [19, §2.4], [20,Corollary 4.2.8], [3] and [2]. However a general solution does not exist.…”

“…This conjecture is known to be true if we replace A n R by an abelian scheme, or, more in general, for smooth projective schemes X (with some extra assumptions) over R provided we consider the abelianization π ab (X, x) of the fundamental group scheme π(X, x) (cf. [3]).…”