On the abelian fundamental group scheme of a family of varieties |
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Authors: | Marco Antei |
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Institution: | 1.Hausdorff Center for Mathematics,Bonn,Germany |
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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
, 0AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{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
X/S
τ
)⋎ → π
1(X, x)
ab
→ π
1(Alb
X/S
, 0AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{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 X
η
of X can be extended to a finite and commutative pointed torsor over X. |
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Keywords: | |
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