Let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$ of characteristic $p>0$. Given a finite commutative group scheme $G$ over $K$ and a smooth projective curve $C$ over $K$ with a rational point, we study the extension of pointed fppf $G$-torsors over $C$ to pointed torsors over some $R$-regular model $\mathcal{C}$ of $C$. We first study this problem in the category of log schemes: given a finite flat $R$-group scheme $\mathcal{G}$, we prove that the data of a pointed $\mathcal{G}$-log torsor over $\mathcal{C}$ is equivalent to that of a morphism $\mathcal{G}^D \to \mathrm{Pic}^{log}_{\mathcal{C}/R}$, where $\mathcal{G}^D$ is the Cartier dual of $\mathcal{G}$ and $\mathrm{Pic}^{log}_{\mathcal{C}/R}$ the log Picard functor. Then, we deduce a criterion for the extension of torsors: it suffices to find a finite flat model of $G$ over $R$ for which a certain group scheme morphism to the Jacobian $J$ of $C$ extends to the Néron model of $J$. In this context, we compute the obstruction for the extended log torsor to come from an fppf one. In a second part, we generalize a result of Chiodo which gives a criterion for the $r$-torsion subgroup of the Néron model of $J$ to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give two detailed examples of extension of torsors when $C$ is a hyperelliptic curve defined over $\mathbb{Q}$, which will illustrates our techniques.
We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for $PGL_p$-torsors on regular noetherian schemes when $p$ is a prime. Extending our previous work when $p=2$, we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all $PGL_p$. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.
Let $X$ be a smooth projective curve over the complex numbers. We compute the Brauer group of the moduli stack of Bruhat-Tits group scheme $\mathcal{G}$-torsors on $X$. When $g(X) \geq 3$ we compute the Brauer group of the regularly stable locus of the coarse moduli space of semi-stable $\mathcal{G}$-torsors.
In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible nodal curve} and the second one is the construction of a flat degeneration of the moduli space of semistable $G$-torsors when the smooth curve degenerates to an irreducible nodal curve. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an ap…
Let $R$ be a regular semilocal integral domain containing an infinite field $k$. Let $f\in R$ be an element such that for all maximal ideals $\mathfrak m$ of $R$ we have $f\notin\mathfrak m^2$. Let $\mathbf G$ be a reductive group scheme over $R$. Under an isotropy assumption on $\mathbf G$ we show that a $\mathbf G$-torsor over the localization $R_f$ is trivial, provided it is rationally trivial. We show that it is not true without the isotropy assumption. Finally, if $\mathbf G$ is a commutative group scheme of multiplicative type and the regular semilocal ring contains a field of characteri…
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 specia…
We prove the equivalence of three "points of view" of the notion of a G-torsor when the base scheme is a Dedekind scheme. As an application, we show that the fibered category of G-torsors on a regular proper curve over a field k is an Artin stack locally of finite presentation over k.
After recalling some basic facts about F-wound commutative unipotent algebraic groups over an imperfect field F we study their regular integral models over Dedekind schemes of positive characteristic and compute the group of isomorphisms classes of torsors of one-dimensional groups.
Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the étale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of étale …
Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the Néron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for…
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 $X'$ su…
In his proof of the fundamental lemma, Ngô established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant vanishing statement for torsors over loop groups $R((t))$ from a general formula for $\mathrm{Pic}(R((t)))$. In the build up to the product formula, we present general algebraization, approximation, and invariance under Henselian pairs results for torsors, give short new proofs for the Elkik approximation theorem and the Chevalley isomorphism $\…
We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and étale group schemes and we prove that, if $G$ is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of $G$-covers is reducible.
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 regu…
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 fund…
This article concerns the geometry of torsors under an elliptic curve. Let $\OO_K$ be a complete discrete valuation ring with algebraically closed residue field and function field $K$. Let $π$ be a generator of the maximal ideal of $\OO_K$, and $S=\mathrm{Spec}(\OO_K)$. Suppose that we are given $J_K$ an elliptic curve over $K$, with $J$ the connected component of the $S$-N?ron model of $J_K$. Given $X_K/K$ a torsor of order $d$ under $J_K$, let $X$ be the $S$-minimal regular proper model. Then there is an invertible id?al $\mathcal{I}\subset \OO_K$ such that $\mathcal{I}^{d}=π\OO_X\subset \OO…
We give a streamlined proof of ${\mathbb A}^1$-representability for $G$-torsors under "isotropic" reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in ${\mathbb A}^1$-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups.
Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To do so, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let $A$ be a mixed characteristic DVR, $G$ a reductive group scheme over $A$, $X$ an irreducible smooth projective $A$-scheme, $\mathcal G$ a principal $G$-bundle over $X$. Suppose $\mathcal G$ is generically trivial. We prove that in this case $\mathcal G$ is Zariski locally trivial. This result confirms the conjecture.
We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the "local…
For a reductive group scheme $G$ over a semilocal Dedekind ring $R$ with total ring of fractions $K$, we prove that no nontrivial $G$-torsor trivializes over $K$. This generalizes a result of Nisnevich-Tits, who settled the case when $R$ is local. Their result, in turn, is a special case of a conjecture of Grothendieck-Serre that predicts the same over any regular local ring. With a patching technique and weak approximation in the style of Harder, we reduce to the case when $R$ is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when $G$ is semis…
Grothendieck--Serre for constant reductive group schemes
Institute for Advanced Study in Mathematics of Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
A separable, proper morphism of varieties with geometrically connected fibers induces a homotopy exact sequence relating the étale fundamental groups of source, target and fiber. Extending work of dos Santos, we prove the existence of an analogous homotopy exact sequence for fundamental group schemes classifying regular singular stratified bundles, under the additional assumption that the morphism in question can be (partially) compactified to a log smooth morphism.
Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in terms of twisted forms of standard models with respect to infinitesimal group scheme actions, and not via extrinsic equations. The main new idea is to analyze and exploit moduli for fields of representatives in Cohen's Structure Theorem. The results serve for the understanding of genus-one fibrations over higher-dimensional bases.
We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in étale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how thi…
It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regul…
Let R be a semi-local regular ring containing an infinite perfect field, and let K be the field of fractions of R. Let H be a simple algebraic group of type F_4 over R such that H_K is the automorphism group of a 27-dimensional Jordan algebra which is a first Tits construction. If char K is not 2, this means precisely that the f_3 invariant of H_K is trivial. We prove that if an H-torsor is rationally trivial, then it is trivial over R. This result is a particular case of the Grothendieck-Serre conjecture. It continues the recent series of papers by I. Panin, N.Vavilov and the authors, and com…
Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the …
A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\mathrm{GL}_n(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson's co-simplicial ring of adèles …
