Searcharxiv⌕ Search

arXiv subjects

Marco Antei

Publications and source records attributed to Marco Antei.

17 recordsLinked to original sources
arXiv2020

Extension of torsors and prime to $p$ 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.

math.AG↗
arXiv2019

The pseudo-fundamental group-scheme

Let $X$ be any scheme defined over a Dedekind scheme $S$ with a given section $x\in X(S)$. We prove the existence of a pro-finite $S$-group scheme $\aleph(X,x)$ and a universal $\aleph(X,x)$-torsor dominating all the pro-finite pointed torsors over $X$. Though $\aleph(X,x)$ may not be unique in general it still can provide useful information in order to better understand $X$. In a similar way we prove the existence of a pro-algebraic $S$-group scheme $\aleph^{\rm alg}(X,x)$ and a $\aleph^{\rm alg}(X,x)$-torsor dominating all the pro-algebraic and affine pointed torsors over $X$. The case where $X\to S$ has no sections is also considered.

math.AG↗
arXiv2018

Models of torsors over affine spaces

Let $X:=\mathbb{A}^{n}_{R}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}^{n}_{K}$ under the action of an affine finite type group scheme can be extended to a torsor over $\mathbb{A}^{n}_{R}$ possibly after pulling $Y$ back over an automorphism of $\mathbb{A}^{n}_{K}$. The proof is effective. Other cases, including $X=α_{p,R}$, will also be discussed.

math.AG↗
arXiv2017

Towers of torsors over a field

Let $X$ be a projective, connected and smooth scheme defined over an algebraically closed field $k$. In this paper we prove that a tower of finite torsors (i.e., under the action of finite $k$-group schemes) can be dominated by a single finite torsor. Let $G$ be any finite $k$-group scheme and $Y$ any $G$--torsor over $X$ pointed in $y\,\in\, Y(k)$; we define over $Y$, which may not be reduced, in a very natural way the categories of Nori-semistable and essentially finite vector bundles. These categories are proved to be Tannakian. Their Galois $k$-group schemes $π^S(Y,\,y)$ and $π^N(Y,\,y)$, respectively, thus generalize the $S$--fundamental and the Nori fundamental group schemes. The latter still classifies all the finite torsors over $Y$, pointed over $y$. We also prove that they fit in short exact sequences involving $π^S(X,\,x)$ and $π^N(X,\, x)$ respectively, where $x$ is the image of $y$.

math.AG↗
arXiv2016

Models of torsors over curves

Let $R$ be a complete discrete valuation ring with fraction field $K$ and with algebraically closed residue field. Let $X$ be a faithfully flat $R$-scheme of finite type of relative dimension 1 and $G$ be any affine $K$-group scheme of finite type. We prove that every $G$-torsor $Y$ over the generic fibre $X_η$ of $X$ can be extended to a torsor over ${X'}$ under the action of an affine and flat $K$-group scheme of finite type $G'$ where $X'$ is obtained by $X$ after a finite number of Néron blowing ups. Moreover if $G$ is finite and étale (resp. admits a finite and flat model) we find $X'$ such that $G'$ is finite and étale (resp. finite and flat) after, if necessary, extending scalars. We provide examples explaining the new techniques.

math.AG↗
arXiv2016

Models of torsors and the fundamental group scheme

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X \to S$ this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_η$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T \to X_η$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question to find a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We obtain partial answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension $1$, such a model exists on some model of $X_η$ obtained by performing a finite number of Néron blow-ups along a closed subset of the special fiber of $X$. In the first part we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a tannakian category constructed starting from the universal torsor.

math.AG↗
arXiv2015

On the bumpy fundamental group scheme

In this short paper we first recall the definition and the construction of the fundamental group scheme of a scheme $X$ in the known cases: when it is defined over a field and when it is defined over a Dedekind scheme. It classifies all the finite (or quasi-finite) fpqc torsors over $X$. When $X$ is defined over a noetherian regular scheme $S$ of any dimension we do not know if such an object can be constructed. This is why we introduce a new category, containing the fpqc torsors, whose objects are torsors for a new topology. We prove that this new category is cofiltered thus generating a fundamental group scheme over $S$, said \textit{bumpy} as it may not be flat in general. We prove that it is flat when $S$ is a Dedekind scheme, thus coinciding with the \textit{classical} one.

math.AG↗
arXiv2015

On the fundamental group scheme of rationally chain connected varieties

Let $k$ be an algebraically closed field. Chambert-Loir proved that the étale 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 étale. In particular, the fundamental group scheme of a Fano variety is finite and étale.

math.AG↗
arXiv2015

Sur l'existence du sch\'ema en groupes fondamental

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group.

math.AG↗
arXiv2012

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 $R$, $φ_1:G\to G_1$ any morphism of $R$-group schemes and $φ_2:G\to G_2$ 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 $φ_1$ 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.

math.AG↗
arXiv2011

On the Grothendieck-Lefschetz Theorem for a Family of Varieties

Let $k$ be an algebraically closed field of characteristic $p>0$, $W$ the ring of Witt vectors over $k$ and ${R}$ the integral closure of $W$ in the algebraic closure ${\bar{K}}$ of $K:=Frac(W)$; let moreover $X$ be a smooth, connected and projective scheme over $W$ and $H$ a relatively very ample line bundle over $X$. We prove that when $dim(X/{W})\geq 2$ there exists an integer $d_0$, depending only on $X$, such that for any $d\geq d_0$, any $Y\in |H^{\otimes d}|$ connected and smooth over ${W}$ and any $y\in Y({W})$ the natural ${R}$-morphism of fundamental group schemes $π_1(Y_R,y_R)\to π_1(X_R,y_R)$ is faithfully flat, $X_R$, $Y_R$, $y_R$ being respectively the pull back of $X$, $Y$, $y$ over $Spec(R)$. If moreover $dim(X/{W})\geq 3$ then there exists an integer $d_1$, depending only on $X$, such that for any $d\geq d_1$, any $Y\in |H^{\otimes d}|$ connected and smooth over ${W}$ and any section $y\in Y({W})$ the morphism $π_1(Y_R,y_R)\to π_1(X_R,y_R)$ is an isomorphism.

math.AG↗
arXiv2011

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.

math.AG↗
arXiv2010

Galois Closure of Essentially Finite Morphisms

Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$ dominating $f$ by an $X$-morphism $λ:\hat{X}_Y\to Y$ and universal for this property. Moreover we show that $λ:\hat{X}_Y\to Y$ is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over $Y$ is still an essentially finite vector bundle over $X$. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor $f:Y \to X$ under a finite group scheme satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$, $Y$ has a fundamental group scheme $π_1 (Y,y)$ fitting in a short exact sequence with $π_1 (X,x)$.

math.AG↗
arXiv2010

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 $X_η$ 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 $X_s$.

math.AG↗
arXiv2010

On the Abelian Fundamental Group Scheme of a Family of Varities

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 $π_1(X,x)^{ab}\to π_1(\mathbf{Alb}_{X/S},0_{\mathbf{Alb}_{X/S}})$ 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 $0\to (\mathbf{NS}^τ_{X/S})^{\vee}\to π_1(X,x)^{ab}\to π_1(\mathbf{Alb}_{X/S},0_{\mathbf{Alb}_{X/S}}) \to 0$ 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 $X_η$ of $X$ can be extended to a finite and commutative pointed torsor over $X$.

math.AG↗
arXiv2010

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

We show that the natural morphism $ϕ:π_1(X_η,x_η)\to π_1(X,x)_η$ between the fundamental group scheme of the generic fiber $X_η$ 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 $X_η$ to be extended over $X$. We finally provide examples where $ϕ:π_1(X_η,x_η)\to π_1(X,x)_η$ is an isomorphism..

math.AG↗