On the Abelian Fundamental Group Scheme of a Family of Varities

Details On the Abelian Fundamental Group Scheme of a Family of Varities On the Abelian Fundamental Group Scheme of a Family of Varities

2009-12-29

arxiv.org/abs/0912.5319v2

Israel Journal of Mathematics (2010)

Mathematics

Algebraic Geometry

18 pages, minor changes, final version

Scientific paper

Let $S$ be a connected Dedekind scheme and $X$ an $S$-scheme provided with a section $x$. We prove that the morphism of fundamental group schemes $\pi_1(X,x)^{ab}\to \pi_1(\mathbf{Alb}_{X/S},0_{\mathbf{Alb}_{X/S}})$ induced by the canonical morphism from $X$ to its Albanese scheme $\mathbf{Alb}_{X/S}$ (when the latter exists) fits in an exact sequence of group schemes $0\to (\mathbf{NS}^{\tau}_{X/S})^{\vee}\to \pi_1(X,x)^{ab}\to \pi_1(\mathbf{Alb}_{X/S},0_{\mathbf{Alb}_{X/S}}) \to 0$ 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 $X_{\eta}$ of $X$ can be extended to a finite and commutative pointed torsor over $X$.

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