Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

We show that the natural morphism $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ 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 $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ is an isomorphism..Comment: 19 pages, final versio

On the fundamental group scheme of rationally chain connected varieties

Let $k$ be an algebraically closed field. Chambert-Loir proved that the \'etale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain connected variety over $k$ is finite and \'etale. In particular, the fundamental group scheme of a Fano variety is finite and \'etale.Comment: Final version in International Mathematics Research Notices, 201

Extention of Finite Solvable Torsors over a Curve

Let $R$ be a discrete valuation ring with fraction field $K$ and with algebraically closed residue field of positive characteristic $p$. Let $X$ be a smooth fibered surface over $R$ with geometrically connected fibers endowed with a section $x\in X(R)$. Let $G$ be a finite solvable $K$-group scheme and assume that either $|G|=p^n$ or $G$ has a normal series of length 2. We prove that every quotient pointed $G$-torsor over the generic fiber $X_{\eta}$ of $X$ can be extended to a torsor over $X$ after eventually extending scalars and after eventually blowing up $X$ at a closed subscheme of its special fiber $X_s$.Comment: 16 page

Extension of torsors and prime to pp fundamental group scheme

Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_{\eta}$. Let $f: Y \rightarrow X_{\eta}$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_{\eta}$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $\psi^{(p')}: \pi(X_{\eta},x_{\eta})^{(p')}\to \pi(X,x)_{\eta}^{(p')}$ between the prime-to-$p$ fundamental group scheme of $X_{\eta}$ and the generic fibre of the prime-to-$p$ fundamental group scheme of $X$ is an isomorphism. This generalizes a well known result for the \'etale fundamental group. The methods used are purely tannakian.Comment: To appear in Annals de l'Institut Fourie