Related papers: Tori in the Cremona groups

We classify Jordan $G$-tori, where $G$ is any torsion-free abelian group. Using the Zelmanov prime structure theorem, such a class divides into three types, namely, {the Hermitian type, the Clifford type and the Albert type.} We concretely…

In the first part of this article, we answer a question of I. Dolgachev, which is related to the following problem: given a birational map $f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map…

We prove that in a connected group of finite Morley rank the centralizers of decent tori are connected. We then apply this result to the analysis of minimal connected simple groups of finite Morley rank. Our applications include general…

Normal and composition series of groups enumerated by ordinal numbers are studied. The Jordan-Holder theorem for them is proved.

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…

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski…

The aim of this paper is to give a finer geometric description of the algebraic varieties parametrizing conjugacy classes of nonsolvable subgroups in the plane Cremona group.

We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of…

We describe the conjugacy classes of affine automorphisms in the group $Aut(n,\K)$ (respectively $Bir(\K^n)$) of automorphisms (respectively of birational maps) of $\K^n$. From this we deduce also the classification of conjugacy classes of…

We show that every automorphism of the group $\mathcal{G}_n:= \textrm{Aut}(\mathbb{A}^n)$ of polynomial automorphisms of complex affine $n$-space $\mathbb{A}^n=\mathbb{C}^n$ is inner up to field automorphisms when restricted to the subgroup…

We give a complete solution of the linearization problem in the plane Cremona group over an algebraically closed field of characteristic zero.

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up…

This paper contains a new proof of the classification of elements of prime order in the Cremona group Bir(P^2), up to conjugation. In addition, we give explicit geometric constructions of these Cremona transformations, and provide a…

That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible…

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an…

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety…

It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We…

Let $U$ be a smooth and connected curve over an algebraically closed field of positive characteristic, with smooth compactification $X$. We generalize classical Geometric Class Field theory to provide a classification of fppf $G$-torsors…

We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant…