We show that the natural morphism $\varphi :{\pi }_1(X_{\eta },x_{\eta })\to {\pi }_1{(X,x)}_{\eta }$ between the fundamental group scheme of the generic fiber $X_{\eta }$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of $X$ is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed $G$-torsor over $X_{\eta }$ to be extended over $X$. We finally provide examples where $\varphi :{\pi }_1(X_{\eta },x_{\eta })\to {\pi }_1{(X,x)}_{\eta }$ is an isomorphism.

We consider four approaches to relative Gromov–Witten theory and Gromov–Witten theory of degenerations: J. Li’s original approach, B. Kim’s logarithmic expansions, Abramovich–Fantechi’s orbifold expansions, and a logarithmic theory without expansions due to Gross–Siebert and Abramovich–Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov–Witten invariants associated...