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We study algebraicity and smoothness of fixed point stacks for flat group schemes which have a finite composition series whose factors are either reductive or proper, flat, finitely presented, acting on algebraic stacks with affine, finitely presented diagonal. For this, we extend some theorems of [SGA3.2] on functors of homomorphisms Hom(G, H) and functors of reductive subgroups Sub(H) for an affine, possibly non-flat group scheme H., Comment: Added ''main ideas of the proofs'' in the introduction and corrected typos

We prove that if a group scheme of multiplicative type acts on an algebraic stack with affine, finitely presented diagonal then the stack of fixed points is algebraic. For this, we extend two theorems of [SGA3.2] on functors of subgroups of multiplicative type, and functors of homomorphisms from a group of multiplicative type.

We define dilatations of general schemes and study their basic properties. Dilatations of group schemes are -- in favorable cases -- again group schemes, called N\'eron blowups. We give two applications to their cohomology in degree zero (integral points) and degree one (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of $G$ with the graded pieces in its Lie algebra $\mathfrak g$, and we show that many level structures on moduli stacks of $G$-bundles are encoded in torsors under N\'eron blowups of $G$., Comment: This is the joint work of two earlier works on dilatations, conducted around the same time independently by the authors. This new version includes minor corrections

Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms $\sigma_i:x_i\to \mbox{F}^*x_{i+1}$. Let $\mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $\mathscr{X}\to \Pi_1(\mathscr{X}/S)$ the \'etale fundamental pro-groupoid, constructed in the present text. We prove that if $\mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $\mathscr{X}\to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks \[\mathscr{H}\!om(\Pi_1(\mathscr{X}/S),\mathscr{M}) \simeq \mathscr{H}\!om(\mathscr{X},\mbox{Fdiv}(\mathscr{M})).\] In particular, the system of relative Frobenius morphisms $\mathscr{X}\to \mathscr{X}^{p/S}\to \mathscr{X}^{p^2/S}\to\dots$ allows to recover the space of connected components $\pi_0(\mathscr{X}/S)$ and the relative \'etale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$., Comment: 50 pages. Improved the main result and fixed some proofs in section 5. Comments are welcome

Among all affine, flat, finitely presented group schemes, we focus on those that are pure, this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base ring, pure group schemes have a classifying space satisfying the resolution property, an embedding into some GLn, a tensor generator for their category of finite type representations, and can be reconstructed from their category of projective finite type representations. In the case of an Artinian base ring, the same is true for all affine, flat, finitely presented group schemes, this answers a question of Conrad. We also prove that quotients of pure groups by closed pure subgroups over an arbitrary base scheme are Zariski-locally quasi-projective. This answers a question of Raynaud, in the case of affine groups. We give various applications., Comment: The article is withdrawn, due to an error in Thm 4.1 that has consequences in several of our results

We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \rightarrow \text{Lie}(G, I) \rightarrow E \rightarrow G \rightarrow 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonn\'e classification for smooth, commutative, unipotent group schemes over $k[I]$., Comment: 38 pages. Accepted in Epiga, final version

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin., Comment: Published version

Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid $R\rightrightarrows X$ with finite stabilizer to be the length of the canonical sequence of the finite map $R\to X\times\_{X/R} X$, where $X/R$ is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient $X\to X/R$ and a theorem of quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of quotients by groupoids.

In this note, we present the theorem of extension of birational group laws in both settings of classical varieties (Weil) and schemes (Artin). We improve slightly the original proof with a more direct construction of the group extension and the systematic use of algebraic spaces, and we discuss the separation properties of the group extension. We also explain the important application to the construction of N\'eron models of abelian varieties. This note grew out of lectures given by Ariane M\'ezard and the second author at the Summer School "Sch\'emas en groupes" held in the CIRM (Luminy) from 29 August to 9 September, 2011., Comment: 26 pages ; to appear in the proceedings of the summer school "Sch\'emas en groupes", Luminy, aug. 29 - sept. 9, 2011

We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity., Comment: 29 pages

Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme \mu_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of \mu_{p^n,K}, in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n < 4. This leads us to conjecture that all finite flat models of \mu_{p^n,K} are Kummer group schemes., Comment: 63 pages, final version, minor changes. To appear in Annales de l'Institut Fourier

We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to \cX$., Comment: 22 pages

For an algebraic stack $\sX$ flat and of finite presentation over a scheme $S$, we introduce various notions of {\em relative connected components} and {\em relative irreducible components}. The main distinction between these notions is whether we require the total space of a relative component to be open or closed in $\sX$. We study the representability of the associated functors of relative components, and give an application to the moduli stack of curves of genus $g$ admitting an action of a fixed finite group $G$., Comment: 28 pages

Let $R$ be a discrete valuation ring with fraction field $K$ and $X$ a flat $R$-scheme. Given a faithful action of a $K$-group scheme $G_K$ over the generic fibre $X_K$, we study models $G$ of $G_K$ acting on $X$. In various situations, we prove that if such a model $G$ exists, then there exists another model $G'$ that acts faithfully on $X$. This model is the schematic closure of $G$ inside the fppf sheaf $Aut_R(X)$; the major difficulty is to prove that it is representable by a scheme. For example, this holds if $X$ is locally of finite type, separated, flat and pure and $G$ is finite flat. Pure schemes (a notion recalled in the text) have many nice properties : in particular, we prove that they are the amalgamated sum of their generic fibre and the family of their finite flat closed subschemes. We also provide versions of our results in the setting of formal schemes., Comment: 29 pages. Supersedes previous preprint "Effective model of a finite group action", arXiv:math/0601639

For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W, we show that the Chevalley morphism t/W -> g/G is an isomorphism except for the group Sp_{2n} over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g -> g/G commutes, or does not commute, with base change on S., Comment: 27 pages

To each associative unitary finite-dimensional algebra over a normal base, we associative a canonical multiplicative function called its determinant. We give various properties of this construction, as well as applications to the topology of the moduli stack of n-dimensional algebras., Comment: 14 pages

We give a thorough study of Hurwitz stacks both in Galois and non galois case. The construction is applied to revisit somme classical examples, the stack of stable curves equipped with a level structure, and the stacks of tamely ramified cyclic covers. We exhibit some tautological cohomology classes, and give some universal relations between them. Applications are given to Cornalba-Harris type relations, and to some Hurwitz-Hodge integrals., Comment: 155 pages

Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a flat $R$-scheme of finite type and $G$ a finite flat group scheme acting on $X$ so that $G\_K$ is faithful on the generic fibre $X\_K$. We prove that there is an effective model of $G$ i.e. a finite flat group scheme dominated by $G$, isomorphic to it on the generic fibre, and extending the action of $G\_K$ on $X\_K$ to an action on all of $X$ that is faithful also on the special fibre. It is unique with these properties. We give examples and applications to degenerations of coverings of curves.

We give the basic definitions of group actions on (algebraic) stacks, and prove the existence of fixed points and quotients as (algebraic) stacks., Comment: 11 pages, uses xy-pic diagram package

In this article we study the modular properties of a family of cyclic coverings of P^1 of degree N, in all odd characteristics. We compute the moduli space of the corresponding algebraic stack over Z[1/2], as well as the Picard groups over algebraically closed fields. We put special emphasis on the study of the fibre of the stack at a prime of wild ramification; in particular we show that the moduli space has good reduction at such a prime., Comment: 23 pages; 2 figures; 2nd revised version