Intersections of smooth polar unitals with secants are homeomorphic to spheres

In [5] and [6] we proved that (non-empty) sets of absolute points of smooth polarities, i.e. smooth polar unitals, in smooth projective planes of dimension 2l are smooth submanifolds of the point spaces homeomorphic to spheres of dimension . In this paper we show that the intersections of smooth polar unitals with secants are homeomorphic to spheres of dimension , respectively. Furthermore we prove that the condition of connectedness in [6, Theorem 1.2] may be omitted. This means that a closed (not necessarily connected) submanifold U of the point space of a smooth projective plane is homeomorphic to a sphere provided that there exists precisely one tangent at each point of U, and each secant intersects U transversally. If U has codimension 1 in the point space then the second condition follows from the first one, and also the intersections of U with secants are homeomorphic to spheres. This result may be generalized to compact hypersurfaces in the point spaces of smooth affine planes. Received: 1 July 2008     

Hyperbolic unitals in the Hall planes

A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.Work performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T.     

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.     

Non-classical polar unitals in finite Figueroa planes

The finite Figueroa planes are non-Desarguesian projective planes of order q ³ for all prime powers q > 2. These planes were constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundh?fer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Using the result of O??Nan in 1971 on the non-existence of his configuration in a classical unital, and the intrinsic characterization by Taylor in 1974 of the notion of perpendicularity induced by a unitary polarity in the classical plane (introduced by Dembowski and Hughes in 1965), we show that these Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.     

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.     

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.     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

16-Dimensional Smooth Projective Planes with Large Collineation Groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d 39 is isomorphic to the octonion projective plane P₂ O. For topological compact projective planes this is true if d 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.     

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.     

Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ? → ? into a continuous planar function ?² → ?². We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.     

A characterization of certain 3-dimensional affino-projective spaces

The finite planar spaces containing at least one pair of planes intersecting in exactly one point and in which for every such pair of planes Π and Π′, any line intersecting Π intersects Π′ (or is contained in Π) are completely classified. These spaces are essentially obtained from projective spaces PG(3, k) by deleting either k collinear points or an affino-projective (but not projective) plane of order k.     

On regular {v,n}-arcs in finite projective spaces

A regular {v, n}-arc of a projective space P of order q is a set S of v points such that each line of P has exactly 0,1 or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n}-arc with n ≥ √q + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v, n}-arc with n ≥ √q + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U, an affine space of order q in U, or S equals the point set Of U. © 1993 John Wiley & Sons, Inc.     

Constructions of unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital U in . The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of with the following three properties:

(i)

points of Π are those of ;

(ii)

lines of Π are certain lines and conics of ;

(iii)

the points in U form a non-classical B-M unital in Π.

Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.     

Polaritäten Ebener projektiver Ebenen

A projective plane is called flat if the spaces of points and lines are locally compact and 2-dimensional and the joining of points and the intersecting of lines are continuous. H. Salzmann studied planes of this type in [11]–[21]. Here polarities of such planes are considered. In II general properties of polarities of flat planes are discussed. For example, a polarity with absolute points has always an oval of absolute points. A flat projective plane with a cartesian ternary field K admits a polarity iff multiplication in K is commutative. In III the polarities of flat projective planes with a 3-dimensional collineation group are determined.     

Einbettung von topologischen Raumgeometrien auf R3 in den reellen affinen Raum

Betten [1] had defined topological spatial geometries on R ³: In R ³ a system L of closed subsets homeomorphic to R (the lines) and a system ? of closed subsets homeomorphic to R ² (the planes) are given such that through any two different points passes exactly one line and through any three non-collinear points passes exactly one plane. Furthermore, ? and ? carry topologies such that the operations of joining and intersection are continuous. It is proved that any topological spatial geometry on R ³ can be imbedded into R ³ as an open convex subset K such that the lines in ? (planes in ?) are mapped onto intersections of lines (planes) of R ³ with K. The collineation group of the geometry is isomorphic to the subgroup of the colineation group of real projective space consisting of the automorphisms that map K into itself. In particular, it is a Lie group of dimension ?12.     

Unitals in the regular nearfield planes

Classes of parabolic unitals in the regular nearfield planes of odd square order are enumerated and classified. These unitals correspond to certain Buekenhout-Metz unitals in the classical plane. Their collineation groups are determined and the unitals are sorted by projective equivalence.      

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P ₂\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp.     

Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

Let γ be a smooth generic curve in ?P ₃. Denote by C the number of its flattening points, and by T the number of planes tangent to γ at three distinct points. Consider the osculating planes to γ at the flattening points. Let N denote the total number of points where γ intersects these osculating plane transversally. Then T ≡ [N + θ(γ)C]/2 (mod 2), where θ(γ) is the number of noncontractible components of γ. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in ?³ has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.     

An exotic involution of S4

CAPPELL and Shaneson [1] construct a family of smooth 4-manifolds which are simple homotopy equivalent to real projective 4-space RP⁴, but not even smoothly h- cobordant to RP⁴. (It is possible they are homeomorphic to RP⁴.) It is natural to ask whether their double covers are S⁴ or not.