Related papers: Icosahedron in birational geometry
In this paper we describe a group theoretical approach to the study of structural transitions of icosahedral quasicrystals and point arrays. We apply the concept of Schur rotations, originally proposed by Kramer, to the case of aperiodic…
We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes…
A priori, the set of birational transformations of an algebraic variety is just a group. We survey the possible algebraic structures that we may add to it, using in particular parametrised family of birational transformations.
Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and…
Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…
We develop the fundamental theory to study cubical isometry groups as totally disconnected, locally compact groups. We show how cubical isometries are determined by their local actions and how this can be applied in explicit constructions.…
It is well known that the point group of the root lattice D_6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H_3, its roots and weights are determined in terms of those of D_6. Platonic and…
The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…
We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.
We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.
Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every…
We construct a birational equivalence between certain quotients of s-tuples of equidimensional linear subspaces of $C^n$ and some quotients of products of square matrices modulo diagonal conjugations. In particular, we prove the rationality…
We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.
Observations on rational Chow groups and cycle class maps in equivariant contexts.
We prove isoperimetric inequalities for quotients of $n$-dimensional Affine buildings. We use these inequalities to prove topological overlapping for the 2-dimensional skeletons of these buildings.
The quotients of a (non-orientable) quantum Seifert manifold by circle actions are described. In this way quantum weighted real projective spaces that include the quantum disc and the quantum real projective space as special cases are…
Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the…
We consider the incidence structure formed by the twelve pentagons given by the vertex neighborhoods of the icosahedron. Interpreting this structure purely in terms of coplanarity conditions, we show that -- up to projective equivalence --…