The holomorphic automorphism group of the complex disk

The group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex Plane, which are associative-commutative operations forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations forming agyrogroup.     

The automorphism group of a split metacyclic p-group

This paper finds the order, structure and presentation for the automorphism group of a split metacyclic p-group, where p is odd. Received: 8 February 2006     

Set-theoretic defining equations of the tangential variety of the Segre variety

We prove a set-theoretic version of the Landsberg-Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture, we use a connection to the author’s previous work and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than 1. We discuss applications to semiseparable matrices, tensor rank versus border rank, context-specific independence models and factor analysis models.     

Equivariant deformations of the affine multicone over a flag variety

We prove that the invariant Hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. More specifically, we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well-determined wonderful variety.     

Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

On the degree of the inverse of an automorphism of the affine space

We study the degree of the inverse of an automorphism f of the affine n-space over a -algebra k, in terms of the degree d of f and of other data. For n = 1, we obtain a sharp upper bound for deg (f^{− 1}) in terms of d and of the nilpotency index of the ideal generated by the coefficients of f^′'. For n = 2 and arbitrary d≥ 3, we construct a (triangular) automorphism f of Jacobian one such that deg(f^{− 1}) > d. This answers a question of A. van den Essen (see [3]) and enables us to deduce that some schemes introduced by authors to study the Jacobian conjecture are not reduced. Still for n = 2, we give an upper bound for deg (f^{− 1}) when f is triangular. Finally, in the case d = 2 and any n, we complete a result of G. Meisters and C. Olech and use it to give the sharp bound for the degree of the inverse of a quadratic automorphism, with Jacobian one, of the affine 3-space.     

The automorphism group of a split metacyclic 2-group

This paper finds the order, structure and presentation for the automorphism group of any split metacyclic 2-group. Received: 2 August 2006     

A bound for the Castelnuovo-Mumford regularity of log canonical varieties

In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal I in terms of the degrees of its generators. We assume that I defines a local complete intersection with log canonical singularities.     

The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

The relative Brauer group and generalized cross products for a cyclic covering of affine space

We study algebras and divisors on a normal affine hypersurface defined by an equation of the form zⁿ=f(_x1,…,_xm)$z^n=f(x_1,\dots ,x_m)$. The coordinate ring is T=k[_x1,…,_xm,z]/(zⁿ−f)$T=k[x_1,\dots ,x_m,z]/(z^n-f)$, and if R=k[_x1,…,_xm][f⁻¹]$R=k[x_1,\dots ,x_m]\left[f^{-1}\right]$ and S=R[z]/(zⁿ−f)$S=R\left[z\right]/(z^n-f)$, then S$S$ is a cyclic Galois extension of R$R$. We show that if the Galois group is G$G$, the natural map H¹(G,Cl(T))→H¹(G,Pic(S))$H^1(G,Cl\left(T\right))\to H^1(G,Pic\left(S\right))$ factors through the relative Brauer group B(S/R)$B(S/R)$ and that all of the maps are onto. Sufficient conditions are given for H¹(G,Cl(T))$H^1(G,Cl\left(T\right))$ to be isomorphic to B(S/R)$B(S/R)$. As an example, all of the groups, maps, divisors and algebras are computed for an affine surface defined by an equation of the form zⁿ=(y−_a1x)?(y−_anx)(x−1)$z^n=(y-a_{1}x)?(y-a_{n}x)(x-1)$.     

Counterexamples regarding symmetric tensors and divided powers

We investigate the similarities and differences between the module of symmetric tensors and the module of divided powers . There is a canonical map which is an isomorphism in many important cases. We give examples showing that this map need neither be surjective nor injective in general. These examples also show that the functor does not in general commute with base change.     

A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ? [p^?1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.     

Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring $k[X,Y,Z]$ over a field k of characteristic zero extend to the polynomial ring $R[X,Y,Z]$ over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on $k[X_1,X_2,X_3,X_4]$ of rank at most three is a polynomial ring over k.     

A note on residual variables of an affine fibration

In a recent paper [7], M.E. Kahoui has shown that if R   is a polynomial ring over C$\mathbb{C}$, A   an A³${\mathbb{A}}^3$-fibration over R, and W a residual variable of A then A   is stably polynomial over R[W]$R[W]$. In this article we show that the above result holds over any Noetherian domain R   provided the module of differentials _ΩR(A)${\Omega }_R(A)$ of the affine fibration A (which is necessarily a projective A-module by a theorem of Asanuma) is a stably free A-module.     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

The relative Brauer group of a cyclic cover of affine space

We study the finite-dimensional central division algebras over the rational function field in several variables over an algebraically closed field. We describe the division algebras that are split by the cyclic covering obtained by adjoining the nth root of a polynomial. The relative Brauer group is described in terms of the Picard group of the cyclic covering and its Galois group. Many examples are given and in most cases division algebras are presented that represent generators of the relative Brauer group.     

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit of N is contained in the orbit closure and has codimension two. We show that the pointed variety is smoothly equivalent to a simple surface singularity of type A _n , or to the cone over a rational normal curve.     

The vanishing ideal of a finite set of closed points in affine space

Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger-Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.