Pushout of quasi-finite and flat group schemes over a Dedekind ring

Let $G$, $G_1$ and $G_2$ be quasi-finite and flat group schemes over a complete discrete valuation ring $R$, $\varphi_1:G\to G_1$ any morphism of $R$-group schemes and $\varphi_2:G\to G_2$ a model map. We construct the pushout $P$ of $G_1$ and $G_2$ over $G$ in the category of $R$-affine group schemes. In particular when $\varphi_1$ is a model map too we show that $P$ is still a model of the generic fibre of $G$. We also provide a short proof for the existence of cokernels and quotients of finite and flat group schemes over any Dedekind ring.Comment: 18 pages, preliminary versio

The Fundamental Group Scheme of a non Reduced Scheme

We extend the definition of fundamental group scheme to non reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.Comment: Final version, 11 pages, minor change

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