arXiv subjects
Publications and source records attributed to Hans Havlicek.
We show that each Jordan homomorphism $R\to R'$ of rings gives rise to a harmonic mapping of one connected component of the projective line over $R$ into the projective line over $R'$. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over $R$.
The set $G$ of all $m$-dimensional subspaces of a $2m$-dimensional vector space $V$ is endowed with two relations, complementarity and adjacency. We consider bijections from $G$ onto $G'$, where $G'$ arises from a $2m'$-dimensional vector space $V'$. If such a bijection $ϕ$ and its inverse leave one of the relations from above invariant, then also the other. In case $m\geq 2$ this yields that $ϕ$ is induced by a semilinear bijection from $V$ or from the dual space of $V$ onto $V'$. As far as possible, we include also the infinite-dimensional case into our considerations.
In the nineties, A.G. Spera introduced a construction principle for divisible designs. Using this method, we get series of divisible designs from finite Laguerre geometries. We show a close connection between some of these divisible designs and divisible designs whose construction was based on a conic in a plane of a 3-dimensional projective space.
We show that the automorphisms of the flag space associated with a 3-dimensional projective space can be characterized as bijections preserving a certain binary relation on the set of flags in both directions. From this we derive that there are no other automorphisms of the flag space than those coming from collineations and dualities of the underlying projective space. Further, for a commutative ground field, we discuss the corresponding flag variety and characterize its group of automorphic collineations.
The main result of the present paper is that the projective line over a ring $R$ is connected with respect to the relation "distant" if, and only if, $R$ is a $GE_2$-ring.
We describe all adjacency preserving bijections of certain products of Grassmann spaces.
It is well known that the three altitudes of a triangle are concurrent at the so-called orthocenter of the triangle. So one might expect that the altitudes of a tetrahedron also meet at a point. However, it was already pointed out in 1827 by the Swiss geometer Jakob Steiner (1796--1863) that the altitudes of a general tetrahedron are mutually skew, for they are generators of an equilateral hyperboloid.
We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.
A $k$-nucleus of a normal rational curve in PG$(n,F)$ is the intersection over all $k$-dimensional osculating subspaces of the curve ($k\in\{-1,0,...,n-1\}$). It is well known that for characteristic zero all nuclei are empty. In case of characteristic $p>0$ and $# F\geq n$ the number of non-zero digits in the representation of $n+1$ in base $p$ equals the number of distinct nuclei. An explicit formula for the dimensions of $k$-nuclei is given for $# F\geq k+1$.
We discuss representations of the projective line over a ring $R$ with 1 in a projective space over some (not necessarily commutative) field $K$. Such a representation is based upon a $(K,R)$-bimodule $U$. The points of the projective line over $R$ are represented by certain subspaces of the projective space $P(K,U\times U)$ that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.
We give a short proof for the existence of the small Witt design which is based on the projective plane of order three with one point deleted.
An elementary geometric proof for the existence of Witt's 5-(12,6,1) design is given.
Let $χ$ be a linear morphism of the product of two projective spaces $PG(n,F)$ and $PG(m,F)$ into a projective space. Let $γ$ be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism $α$ of the product space and a linear morphisms of projective spaces $ϕ$, such that $γϕ=αχ$.
Let $Γ'$ and $Γ$ be two Grassmannians. The standard embedding $ϕ:Γ'\timesΓ\to \bar{P}$ is obtained by combining the Plücker and Segre embeddings. Given a further embedding $η: Γ'\timesΓ\to P'$, we find a sufficient condition for the existence of $α\in Aut(Γ)$ and of a collineation $ψ: \bar{P} \to P'$ such that $η=({\rm id}_{Γ'}\timesα)ϕψ$.
The nucleus of a Veronese variety is the intersection of all its osculating hyperplanes. Various authors have given necessary and sufficient conditions for the nucleus to be empty. We present an explicit formula for the dimension of this nucleus for arbitrary characteristic of the ground field. As a corollary, we obtain a dimension formula for that subspace in the $t$-th symmetric power of a finite-dimensional vector space $V$ which is spanned by the powers $a^t$ with $\a\in\V$.
We introduce the chain geometry $Σ(K,R)$ over a ring $R$ with a distinguished subfield $K$, thus extending the usual concept where $R$ has to be an algebra over $K$. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of $K$ is normal in the group of units of $R$. This condition is not equivalent to $R$ being a $K$-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.
Any set of $σ$-Hermitian matrices of size $n \times n$ over a field with involution $σ$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.