Dimensional Affine Space Research Articles

UNE PROPRIÉTÉ DU GROUPE Á 168 ÉLÉMENTS

Let e be an affine space of dimension n over a field the affine group of e, G the corresponding linear group. To each point corresponds a section of the canonical map to the linear map corresponds the affine map wich has as associated linear map and a as fixed point. We proove that every section is of this type, except in the only one case where K = F2 and n = 3.

On a Normal of the Distribution of Hyperplane Elements in an Affine Space

Using the CartanLaptev invariant analytic method, an invariant affine normal (𝐸) is constructed, which is intrinsically connected with the distribution of hyperplane elements in the (𝑛 1)-dimensional affine space. For the normal (𝐸) we define a second kind normal and an invariant (𝑛 1)-dimensional plane lying in the plane of the element and not passing through the center and corresponding to this normal in the BompianiPantazi projectivity. An invariant point of intersection of the two-dimensional plane passing through the normals (𝐿) and (𝐸) with the second kind normal is found.

Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Suss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.

Restriction of a cuspidal module for finite general linear groups

In this article, we consider two subgroups A and A′ of GL n ( q) containing a fixed unipotent subgroup U, either of which may be identified with the group of all affine transformations of an n−1 dimensional affine space. We describe the restriction of an irreducible cuspidal module to the subgroup A∩ A′ of GL n ( q) and prove that it is multiplicity-free.

On holomorphic one-forms transverse to closed hypersurfaces

In this note we announce some achievements in the study of holomorphic distributions admitting transverse closed real hypersurfaces. We consider a domain with smooth boundary in the complex affine space of dimension two or greater. Assume that the domain satisfies some cohomology triviality hypothesis (for instance, if the domain is a ball). We prove that if a holomorphic one form in a neighborhood of the domain is such that the corresponding holomorphic distribution is transverse to the boundary of the domain then the Euler-Poincaré-Hopf characteristic of the domain is equal to the sum of indexes of the one-form at its singular points inside the domain. This result has several consequences and applies, for instance, to the study of codimension one holomorphic foliations transverse to spheres.

Parallel affine immersions with maximal codimension

We study affine immersions, as introduced by Nomizu and Pinkall, of $M^n$ into $\R^{n+p}$. We call $M^n$ linearly full if the image of $M$ is not contained in a lower dimensional affine space. Typical examples of affine immersions are the Euclidean and semi-Riemannian immersions. A classification, under an additional assumption that the rank of the second fundamental form is at least two, of the hypersurfaces with parallel second fundamental form was obtained by Nomizu and Pinkall. If we assume that the second fundamental form is parallel and $M$ is linearly full, then $p \le n(n+1)/2$. In this paper we completely classify the affine immersions with parallel second fundamental form in $\R^{n+n(n+1)/2}$, obtaining amongst others the generalized Veronese immersions.

On Surfaces with Prescribed Shape Operator

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. It is shown that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.

On polynomial automorphisms of affine spaces

In the first part of this paper we prove some general results on the linearizability of algebraic group actions on . As an application, we get a method of construction and concrete examples of non-linearizable algebraic actions of infinite non-reductive insoluble algebraic groups on with a fixed point. In the second part we use these general results to prove that every effective algebraic action of a connected reductive algebraic group on the -dimensional affine space over an algebraically closed field of characteristic zero is linearizable in each of the following cases: 1) ; 2) and is not a one- or two-dimensional torus. In particular, this means that is the unique (up to conjugacy) maximal connected reductive subgroup of the automorphism group of the algebra of polynomials in three variables over .

Rationality of moduli of vector bundles on curves

The moduli space M(r,d) of stable, rank r, degree d vector bundles on a smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A, where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The birational isomorphism is compatible with fixing determinants in M(r,d) and M(h,0) and we obtain as a corollary that the moduli space of bundles of rank r and fixed determinant of degree d is rational, when r and d are coprime. A key ingredient in the proof is the use of a naturally defined Brauer class for the function field of M(r,d).

Random nested tetrahedra

In a real n-1 dimensional affine space E, consider a tetrahedron T0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T0, thus obtaining a new tetrahedron T1 contained in T0. Repeat the operation with T1 instead of T0, obtaining T2, and so on. The sequence of the Tk shrinks to a point Y of T0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, αn) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

Random nested tetrahedra

In a real n-1 dimensional affine space E, consider a tetrahedron T 0, i.e. the convex hull of n points α1, α2, …, α n of E. Choose n independent points β1, β2, …, β n randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, α n ) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

On Flat Flag-Transitive c.c^* -Geometries

We study flat flag-transitive c.c^{\ast}-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of then -dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism groupG = 2^{2n}\cdot L_n(2) and covered by the truncated Coxeter complex of typeD_{2^n} . The non-canonical ways give us geometries with smaller automorphism group (G \le 2^{2n}\cdot (2^n-1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.

On a partition of a three dimensional affine space over GF(C), q odd

Abstract By some properties of a special partition in lines of AG(3,q) we shall build a blocking set with respect to the lines of AG(3,q).

Geometry of dots and ropes

An a-dot is the first infinitesimal neighbourhood of a point with respect to an (a 1)-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let r be a collection of d a-dots in uniform position in Pn, a > 2. Then the Hilbert function hr of F satisfies hr(r) ? min(rn + 1, 2d) + (a 2) min((r l)n 1, d) for r > 3. Equality occurs for some r with rn + 2 < 2d if and only if Fred is contained in a rational normal curve C, and the tangent directions to this curve at these points are all contained in F. Equality occurs for some r with (r 1 )n < d if and only if r is contained in thefirst infinitesimal neighbourhood of C with respect to a subbundle, of rank a 1 and of maximal degree, of the normal bundle of C in P'n. This implies an upper bound on the degree of a subbundle of rank a 1 of the normal bundle of an irreducible nondegenerate smooth curve of degree d in P'n , by a Castelnuovo argument.

Geometry of Dots and Ropes

An a-dot is the first infinitesimal neighbourhood of a point with respect to an (a 1)-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let r be a collection of d a-dots in uniform position in Pn, a > 2. Then the Hilbert function hr of F satisfies hr(r) ? min(rn + 1, 2d) + (a 2) min((r l)n 1, d) for r > 3. Equality occurs for some r with rn + 2 < 2d if and only if Fred is contained in a rational normal curve C, and the tangent directions to this curve at these points are all contained in F. Equality occurs for some r with (r 1 )n < d if and only if r is contained in thefirst infinitesimal neighbourhood of C with respect to a subbundle, of rank a 1 and of maximal degree, of the normal bundle of C in P'n. This implies an upper bound on the degree of a subbundle of rank a 1 of the normal bundle of an irreducible nondegenerate smooth curve of degree d in P'n , by a Castelnuovo argument.

Reconstructing birational maps from their face functions

We first prove that any birational map, from an affine space of dimension ≥ 2 to itself, is not determined by its face functions. On the other hand, we prove that a birational map with irreducibly polynomial inverse is completely determined, within the class of all birational maps with irreducibly polynomial inverses, by its face functions. We show also how to effectively reconstruct such a map from its face functions.

Embeddings of Stein Manifolds of Dimension n into the Affine Space of Dimension 3n/2 + 1

Embeddings of Stein Manifolds of Dimension n into the Affine Space of Dimension 3n/2 + 1

Rank n Geometries with Affine Hyperplanes and Dual Affine Point Residues

This paper extends to any rank n a previous classification result [4, 5], on rank 3 geometries with affine planes and dual affine point residues. Let Г be a rank n incidence geometry of points, lines, ..., hyperplanes, the hyperplanes of which are affine spaces, the point residues of which are dual affine spaces and which satisfies Axiom [A]: any two points of Г are incident with at most one line. Such a geometry is necessarily isomorphic to the incidence structure obtained from any n -dimensional affine space A either by deleting a point and all subspaces of A incident with that point or by deleting a direction of lines and all subspaces of A parallel to that direction.

A Classification of Non-Degenerated Binary Forms

There is given a representation of the equivalence classes of non-degenerated binary forms of degree d as quotient of an affine open subset of (d – 3)-dimensional affine space by the d-th symmetric group.

STABILIZATION OF SINGULARITIES OF CONVEX HULLS

Normal forms are found for simple singularities of convex hulls of general compact manifolds of codimension not greater than two in an affine space of dimension not greater than seven.Figures: 3. Bibliography: 5 titles.