On the Abelian Fundamental Group Scheme of a Family of Varities

TL;DR

This paper investigates the structure of the abelian fundamental group scheme of a family of varieties over a Dedekind scheme, establishing an exact sequence involving the Albanese scheme and extending torsors from the generic fiber.

Contribution

It proves an exact sequence relating the abelian fundamental group scheme, the Albanese scheme, and the Néron-Severi group scheme, and shows torsors over the generic fiber extend to the entire family.

Findings

Established an exact sequence of fundamental group schemes.

Proved the kernel is a finite flat group scheme.

Extended torsors from generic fiber to whole family.

Abstract

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 $X$.