Collineation groups preserving a unital in a projective plane of even order

We investigate the structure of a collineation group G leaving invariant a unital U in a finite projective plane of even order n=m ². When G is transitive on the points of U and the socle of G has even order, then must be a Desarguesian plane, U a classical unital and PSU(3,m ²)GPU(3,m ²) — for m>2. The primitive case follows as an easy corollary.This research was supported by a grant from the M.P.I.     

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.     

Collineation groups of derived semifield planes III

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Collineation groups whose order is divisible by a p-primitive divisor

Summary Let be a translation plane of order q³ with kernel GF(q). Our main result is that the translation complement of cannot contain a group G such that G/Z(G)=A₇. This removes a possible exception to the results in our paper Collineation groups irreducible on the components of a translation plane.We also show that the assumptions of the above paper can be relaxed slightly.Both authors supported in part by NSF Grant No. MCS76-06661 A01     

Oka's conjecture on irreducible plane sextics

We partially prove and partially disprove Oka's conjecture onthe fundamental group/Alexander polynomial of an irreducibleplane sextic. Among other results, we enumerate all irreduciblesextics with simple singularities admitting dihedral coveringsand find examples of Alexander equivalent Zariski pairs of irreduciblesextics.     

Collineation groups of the intersection of two classical unitals

Kestenband proved in [12] that there are only seven pairwise non‐isomorphic Hermitian intersections in the desarguesian projective plane PG(2, q) of square order q. His classification is based on the study of the minimal polynomials of the matrices associated with the curves and leads to results of purely combinatorial nature: in fact, two Hermitian intersections from the same class might not be projectively equivalent in PG(2, q) and might have different collineation groups. The projective classification of Hermitian intersections in PG(2, q) is the main goal in this paper. It turns out that each of Kestenband's classes consists of projectively equivalent Hermitian intersections. A complete classification of the linear collineation groups preserving a Hermitian intersection is also given. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 445–459, 2001     

Cubic irreducible graphs for the projective plane

A characterization of all cubic finite graphs that do not embed in the real projective plane P is given in the sense that Kuratowski characterized all non-planar finite graphs. Specifically it is shown that there exist exactly 6 cubic irreducible graphs for P.     

Coset intersection of irreducible plane cubics

In a projective plane $PG(2,\mathbb K )$ over an algebraically closed field $\mathbb K $ of characteristic $p\ge 0$ , let $\Omega $ be a pointset of size $n$ with $5\le n \le 9$ . The coset intersection problem relative to $\Omega $ is to determine the family $\mathbf F$ of irreducible cubics in $PG(2,\mathbb K )$ for which $\Omega $ is a common coset of a subgroup of the additive group $(\mathcal F ,+)$ for every $\mathcal F \in \mathbf F$ . In this paper, a complete solution of this problem is given.     

Rational irreducible plane continua without the fixed-point property

We define rational irreducible continua in the plane that admit fixed-point-free maps with the condition that all of their tranches have the fixed-point property. This answers in the affirmative a question of Hagopian. The construction is based on a special class of spirals that limit on a double Warsaw circle. The closure of each of these spirals has the fixed-point property.