Related papers: Local triviality for G-torsors

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern classes…

Let $X$ be an irreducible smooth projective algebraic curve of genus $g \geq 2$ over the ground field $\bc$ and let $G$ be a semisimple simply connected algebraic group. The aim of this paper is to introduce the notion of semistable and…

Let $\mathrm G$ be an isotrivial reductive group over a scheme $S$. We construct a smooth projective $S$-scheme containing $\mathrm G$ as a fiberwise-dense open subscheme equipped with left and right actions of $\mathrm G$ which extend the…

We generalize Hrushovski's Group Configuration Theorem to quasiminimal classes. As an application, we present Zariski-like structures, a generalization of Zariski geometries, and show that a group can be found there if the pregeometry…

We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies…

Let $G$ be an affine group over a field of characteristic not two. A $G$-torsor is called isotropic if it admits reduction of structure to a proper parabolic subgroup of $G$. This definition generalizes isotropy of affine groups and…

We prove that if the moduli $\mathbb Q$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is $\mathbb Q$-b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is…

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H^1(K, S) -> H^1(K, G) is surjective for every field extension…

We show that a real rational (over $\C$) surfaces are quasi-simple, i.e., that such a surface is determined up to deformation in the class of real surfaces by the topological type of its real structure.

It is conjectured by de Jong that, if $X$ is a connected projective smooth variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial etale fundamental group, any convergent isocrystal $\mathcal{E}$ on $X$ is…

Let C be a hyperelliptic curve of good reduction defined over a discrete valuation field K with algebraically closed residue field k. Assume moreover that char k \ne 2. Given d \in K^*\K^*2, we introduce an explicit description of the…

Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $\delta(C)\in…

Let $R$ be a semilocal geometrically factorial Noetherian domain of characteristic zero. We show that a reductive $R$-group scheme is isotropic if it is generically isotropic. We derive various consequences, in particular for the…

Let R\_n be the ring of Laurent polynomials in n variables over a field k of characteristic zero and let K\_n be its fraction field.Given a linear algebraic k-group $G$, we show that a K\_n-torsor under G which is unramified with respect to…

Let (K,v) be a valued field, Y a K-variety, G an algebraic group over K (not necessarily smooth), and f: X->Y a G-torsor over Y. We consider the induced map X(K)-->Y(K), which is continuous for the topologies deduced from the valuation. Let…

In this paper, we study the ordinarity of an isotrivial elliptic surface defined over a field of positive characteristic. If an isotrivial elliptic fibration $\pi:X \to C$ is given, $X$ is ordinary when the common fiber of $\pi$ is ordinary…

Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated…

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the N\'eron-Severi…

Let X be a geometrically rational (or more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough then X contains many rational curves defined over K. As a consequence we prove that…

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the…