Unitals in the Hall plane

The unitals in the Hall plane are studied by deriving PG(2,q ²)and observing the effect on the unitals of PG(2,q ²).The number of Buekenhout and Buekenhout-Metz unitals in the Hall plane is determined. As a corollary we show that the classical unital is not embeddble in the Hall plane as a Buekenhout unital and that the Buekenhout unitals of H(q ²)are not embeddable as Buekenhout unitals in the Desarguesian plane. Finally, we generalize this technique to other translation planes.     

A group theoretic characterization of Buekenhout-Metz unitals

It is shown that a unital U embedded in PG(2,q²) is a Buekenhout-Metz unital if and only if U admits a linear collineation group that is a semidirect product of a Sylow p-subgroup of order q³ by a subgroup of order q − 1. This is the full linear collineation group of U except for two equivalence classes of unitals: (i) the classical unitals, and (ii) the Buekenhout-Metz unitals which can be expressed as a union of a partial pencil of conics. The unitals in class (ii) only occur when q is odd, and any two of them are projectively equivalent. © 1996 John Wiley & Sons, Inc.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

Unitals in the code of the Hughes plane

A coding‐theoretic characterization of a unital in the Hughes plane is provided, based on and extending the work of Blokhuis, Brouwer, and Wilbrink in PG(2,q²). It is shown that a Frobenius‐invariant unital is contained in the p‐code of the Hughes plane if and only if that unital is projectively equivalent to the Rosati unital. © 2003 Wiley Periodicals, Inc.     

Arcs in the Hall Planes

Using a transformation tecnique for designs introduced in [1], I construct a class of arcs embeddeable in the Hall plane and in the dual of the Hall plane of order q proving also their completeness in the unital of Grüning. Math. Subj. Class.: 51A35 Non Desarguesian affine and projective planes. 51E22 Blocking sets, ovals, k-arcs.     

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.     

Characterizations of Buekenhout-Metz unitals

We give a characterization of the Buekenhout-Metz unitals in PG(2, q ²), in the cases that q is even or q=3, in terms of the secant lines through a single point of the unital. With the addition of extra conditions, we obtain further characterizations of Buekenhout-Metz unitals in PG(2, q ²), for all q. As an application, we show that the dual of a Buekenhout-Metz unital in PG(2, q ²) is a Buekenhout-Metz unital.     

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.This paper was written while the third author was supported by a grant from DFG and TÜBITAK.Received March 12, 2002 Published online November 18, 2002     

Some design-theoretic properties of buekenhout unitals

The purpose of this article is to discuss some questions about parabolic Buekenhout unitals, considered as designs. In this article, we define a parabolic Buekenhout unital to be a unital in any two-dimensional translation plane obtained from the cone over any ovoid. In particular, we discuss resolutions of these designs, inversive plane residuals obtainable from these designs, and also some issues about disjoint Steiner systems. © 1996 John Wiley & Sons, Inc.     

Unitals and Unitary Polarities in Symmetric Designs

We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrary symmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shown that if a unital in a point-hyperplane design PG _d-1(d,q) exists, then d = 2 or 3; in particular, unitals and ovoids are equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as the PG ₄(5,2), although the latter does not have a unital. It had been not known whether or not a nonclassical design exists, which has a unitary polarity. Fortunately, we have discovered a unitary polarity in a symmetric 2-(45,12,3) design. To a certain extent this example seems to be exceptional for designs with these parameters.     

Transformation of incidence structures and sharply multiply transitive permutation sets

In this note we generalize the transformation process introduced in [8]. This generalization allows to find the nonplanar nearfield discovered by H.Karzel and W.Kerby, [5], [6], and the affine Andrè planes.Dedicated to Prof. A.Barlotti and to Prof. L.A.Rosati on their 70th birthdayWork performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T.     

Perturbation d'une matrice hermitienne ou normale

Summary Given a hermitian (normal) matrixA with known eigenelements, we study the behavior of these elements under a hermitian perturbationH. With a symmetric 2×2 matrix, the problem is explicit (algebraic equation of 2nd degree), and we try, in the case of ann×n hermitian matrix, to obtain upper bounds which are as close as possible of exact results forn=2. The results are collected in § I. They state in a unified manner some results of Davis [2], Gavurin [4], Golub [5], Kato-Temple [7], Ortega [3], Wilkinson [8] and others.In § II, we apply this theory to produce error bounds for eigenelements computed by theQR and Jacobi methods. The given error bounds are realistic and easy to compute during the algorithms. When the separation distance between eigenvalues ofA approaches zero, the problem of computing eigenelements ofA+H is ill-conditionned with respect to the eigenvalues, the eigenvectors being orthogonal. The precision then obtained is given.

---

Perturbation d'une matrice hermitienne ou normale

     

Perfect matchings in regular bipartite graphs

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk – 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank – 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr 0, stillG – S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.     

On central collineations of derived semifield planes

LetG denote the collineation group generated by the set of all affine central collineations in a derived semifield plane. We present a characterization of the Hall planes in terms of the order ofG. This essentially allows the extension of the theorems of Kirkpatrick and Rahilly on generalized Hall planes to arbitrary derived semifield planes. That is, a derived semifield plane of order q² is a Hall plane precisely when it admits q+1 involutory central collineations.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q ²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q ²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q ²).     

Polarity and transitive parabolic unitals in translation planes of odd order

A parabolic unital of a translation plane is called transitive, if the collineation group G fixing fixes the point at infinity of and acts transitively on the affine points of . It has been conjectured that if a transitive parabolic unital consists of the absolute points of a unitary polarity in a commutative semi-field plane, then the sharply transitive normal subgroupK of G is not commutative. So far, this has been proved for commutative twisted field planes of odd square order, see [1],[5]. Here we prove this conjecture for commutative Dickson planes. Received 14 May 2001.     

Plane curves whose tangent lines at collinear points are concurrent

We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.     

Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2

The aim of this article is to prove a criterion for projectively Cohen-Macaulay two-codimensional subschemes ofP _k ^N to be smoothable. For curves inP _k ³ this criterion is due to T. Sauer [4]. The considered schemes are determinantal, so we study related generic affine determinantal schemes. We compute their dimension, and under the condition that the dimension is minimal we calculate the codimension of the singular locus.