Related papers: Jordan property for Cremona group over a finite fi…
In this paper we find the exact value of the Jordan constant for Cremona group of rank $2$ over all finite fields. During the proof we construct a cubic surface over $\mathbb{F}_2$ with a regular action of the group $\mathrm{S}_6$ which is…
In this paper we compute the Jordan constants of the Cremona group of rank two over all fields of characteristic zero.
We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks 2 and 3.
We bound the indices of normal abelian subgroups in finite groups contained in the Cremona group of rank 2 over a field of odd characteristic.
We compute the Jordan constant for the group of birational automorphisms of a projective plane $\mathbb{P}^2_{\mathbb k}$, where ${\mathbb k}$ is either an algebraically closed field of characteristic 0, or the field of real numbers, or the…
We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.
We show that the Jordan constant for the volume-preserving plane Cremona group $\mathrm{Bir}(\mathbb P^2, \Delta)$ is $12$. We provide a Jordan bound of $144$ for the three-dimensional volume-preserving Cremona group $\mathrm{Bir}(\mathbb…
The Cremona group of rank n over a field k is the group of birational automorphisms of the n-dimensional projective space over the field k. We study the minimal dimension such that all finite subgroups of the Cremona group have a faithful…
In this paper, we find the maximum values that the Jordan constant of the automorphism group of a smooth two-dimensional rational quadric over a field of characteristic zero can attain, depending on the arithmetic properties of a field.
We compute the exact values of the Jordan constants of abelian surfaces over finite fields.
We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of simple polarized abelian fourfolds over finite fields. As an application, we compute the Jordan constants of…
We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and $p$-Jordan property. In particular, we show that the Cremona group of rank $2$ over a field of…
We prove that the group of birational automorphisms of a geometrically irreducible algebraic surface over a finite field is Jordan. We show that the analogous statement fails in higher dimensions. Finally, we prove that groups of birational…
We give an explicit bound on orders of finite subgroups of Cremona group of rank three over $\mathbb{Q}$.
We determine the Jordan constants of groups $\mathrm{GL}_2(K)$, $\mathrm{SL}_2(K)$, $\mathrm{PGL}_2(K)$ and $\mathrm{PGL}_3(K)$ for any given field $K$ of characteristic 0.
Assuming Borisov--Alexeev--Borisov conjecture, we prove that there is a constant $J=J(n)$ such that for any rationally connected variety $X$ of dimension $n$ and any finite subgroup $G\subset Bir(X)$ there exists a normal abelian subgroup…
By a classical result of Jordan, each finite subgroup G of a complex linear group GL_n(C) has an abelian subgroup whose index in G is bounded by a constant depending only on n. We consider the problem if this remains true for finite…
We obtain a sharp bound for p-elementary subgroups in the plane Cremona group over an arbitrary perfect field.
We compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative…
We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These…