Related papers: A note on certain Tannakian group schemes

Let $k$ be an algebraically closed field. Let $C$ be an irreducible smooth projective curve over $k$. Let $E$ be a locally free sheaf on $C$ of rank $\geq 2$. Fix an integer $d \geq 2$. Let $\mathcal{Q}$ denote the Quot scheme…

We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral tannakian categories.

In this note we generalize Nori's definition of the fundamental group scheme from a rational point to an arbitrary base point so that when we take $X$ to be a field $k$ and the point to be $k\subseteq \bar{k}$ we still get a non trivial…

Let $k$ be a field, $K/k$ a field extension, $X$ a connected scheme proper over $k$, $x_K\in X_K(K)$ lying over $x\in X(k)$, $\mathcal{C}_X$ and $\mathcal{C}_{X_K}$ the Tannakian categories over $X$ and $X_K$ respectively,…

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals…

In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $\Lambda^2 TX$, then it is isomorphic to…

We consider parahoric Bruhat-Tits group schemes over a smooth projective curve and torsors under them. If the characteristic of the ground field is either zero or positive but not too small and the generic fiber is absolutely simple and…

Relying on a notion of "numerical effectiveness" for Higgs bundles, we show that the category of "numerically flat" Higgs vector bundles on a smooth projective variety $X$ is a Tannakian category. We introduce the associated group scheme,…

Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations…

Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field $\textbf{k}_F$ and valuation ring $\mathcal{O}_F$, there exists an integral separated smooth group scheme…

Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the…

Given a smooth family F/Y of geometrically irreducible surfaces, we study sequences of arbitrarily near T-points of F/Y; they generalize the traditional sequences of infinitely near points of a single smooth surface. We distinguish a…

In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is $\mathbb P^2$, then such curves must be of…

We study Fourier-Mukai transforms for smooth projective varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Our results lead to new equivalences of derived…

Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of…

We investigate how the \'etale fundamental group controls local systems in characteristic $p$, namely $F$-divided sheaves. In analogy with Grothendieck-Malcev's results for discrete groups, we show that if a morphism $f \colon Y \to X$ of…

Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line…

Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's \'etale…

We prove a "Generic Equivalence Theorem which says that two affine morphisms $p: S \to Y$ and $q: T \to Y$ of varieties with isomorphic (closed) fibers become isomorphic under a dominant etale base change $\phi: U \to Y$. A special case is…

In this work we study smooth complex quasi-projective surfaces whose fundamental group is a free product of cyclic groups. In particular, we prove the existence of an admissible map from the quasi-projective surface to a smooth complex…