Ovoids with a pencil of translation ovals

We show that if an ovoid of PG(3, q), where q>2 is even, has a pencil of translation ovals and if the carrier of the pencil is not an axis of at least one of the ovals in the pencil, then the ovoid is a Tits ovoid. It follows, as a corollary of this and a result of Penttila and Praeger, that if an ovoid of PG(3, q), where q>2 is even, has a pencil of translation ovals then the ovoid is either an elliptic quadric or a Tits ovoid.     

Some inherited maximal arcs in derived dual translation planes

In 1974 J. A. Thas constructed a class of maximal arcs in certain translation planes of orderq ². In this paper a new class of maximal arcs is constructed in certain derived dual translation planes that are inherited from the duals of the Thas maximal arcs. It is noted that some (but not all) of the maximal arcs are isomorphic to a class constructed by the author.The author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

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.     

Spreads and ovoids in finite generalized quadrangles

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s ²), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3 ^e ),e3, is constructed. Then, by the duality betweenQ(4, 3 ^e ) and the classical generalized quadrangleW (3 ^e ), we get line spreads of PG(3, 3 ^e ) and hence translation planes of order 3^2e . These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q ²,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q ²),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q).     

A characterization of Buekenhout-Metz unitals in PG(2, q 2), q even

The known examples of embedded unitals (i.e. Hermitian arcs) in PG(2, q ²) are B-unitals, i.e. they can be obtained from ovoids of PG(3, q) by a method due to Buekenhout. B-unitals arising from elliptic quadrics are called BM-unitals. Recently, BM-unitals have been classified and their collineation groups have been investigated. A new characterization is given in this paper. We also compute the linear collineation group fixing the B-unital arising from the Segre-Tits ovoid of PG(3, 2 ^r ), r3 odd. It turns out that this group is an Abelian group of order q ².Research supported by MURST.     

Ovoids and monomial ovals

This paper is a contribution to the classification of ovoids. We show, under some rather technical assumptions, that if an ovoid of PG(3, q) has a pencil of monomial ovals, then it is either an elliptic quadric or a Tits ovoid. Further, we show that if an ovoid of PG(3, q) has a bundle of translation ovals, again under some extra assumptions, then the ovoid is an elliptic quadric or a Tits ovoid.     

On the sharpness of a theorem of B. segre

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s ² by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q ³) over GF(q), and on a representation of GF(q)³ in GF(q ³)³ indicated in Jamison [4].     

Cyclic caps in PG(3,q)

This article investigates cyclic completek-caps in PG(3,q). Namely, the different types of completek-capsK in PG(3,q) stabilized by a cyclic projective groupG of orderk, acting regularly on the points ofK, are determined. We show that in PG(3,q),q even, the elliptic quadric is the only cyclic completek-cap. Forq odd, it is shown that besides the elliptic quadric, there also exist cyclick-caps containingk/2 points of two disjoint elliptic quadrics or two disjoint hyperbolic quadrics and that there exist cyclick-caps stabilized by a transitive cyclic groupG fixing precisely one point and one plane of PG(3,q). Concrete examples of such caps, found using AXIOM and CAYLEY, are presented.     

Tubes of even order and flat π.C 2 geometries

Atube of even orderq=2 ^d is a setT={L, } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q ₃ ⁺ (q) in PG(3,q); letL be an exterior line, and let consist of the polar line ofL together with a regulus onQ.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PL(4,q) fixing a tube {L, } cannot act transitively on . As pointed out by a construction due to Pasini, this implies new results for the existence of flat .C ₂ geometries whoseC ₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing hyperoval by conic in the definition), and consider briefly a related extremal problem.Dedicated to luigi antonio rosati on the occasion of his 70th birthday     

Caps with free pairs of points

We say that two points x, y of a cap C form a free pair of points if any plane containing x and y intersects C in at most three points. For given N and q, we denote by m₂⁺ (N, q) the maximum number of points in a cap of PG(N, q) that contains at least one free pair of points. It is straightforward to prove that m₂⁺ (N, q) ≤ (q^N-1 + 2q − 3)/(q − 1), and it is known that this bound is sharp for q = 2 and all N. We use geometric constructions to prove that this bound is sharp for all q when N ≤ 4. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and MITACS NCE of Canada.     

Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q ² is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p ^e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

A characterisation of spreads ovally-derived from Desarguesian spreads

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding ovally-derived planes are generalised André planes, of order 2 ^N , although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq ² can be ovally-derived from a Desarguesian spread if and only ifq is a square.     

Random constructions and density results

In this paper we outline a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG(n, q) and which must have a lot of more applications. We also give a survey on what is known about the spectrum of sizes of maximal partial line spreads in PG(n, q). At the end we list some more elaborate random techniques used in finite geometry.      

How many s-subspaces must miss a point set in PG(d, q)

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG(n, q) with the property that every s-subspace has a point in the set is (q ^n+1-s - 1)/(q - 1). We solve in finite projective spaces PG(n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ (q ^n+1-s - 1)/(q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ (q ^d+1 - 1)/(q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG(n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner.     

On the non-existence of linear codes attaining the Griesmer bound

A lower bound on the size of a set K in PG(3, q) satisfying for any plane of PG(3, q), q4 is given. It induces the non-existence of linear [n,4,n + 1 – q ²]-codes over GF(q) attaining the Griesmer bound for .     

Blocking structures in finite projective planes

Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 3^2h). Also, we show that these blocking sets of PG(2, q²), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 3^2h) gives rise to a family of blocking semiovals in the André plane as well.     

Carleson's counterexample and a scale of Lorentz-BMO spaces on the bitorus

We introduce a full scale of Lorentz-BMO spaces BMO _L ^p,q on the bidisk, and show that these spaces do not coincide for different values ofp andq. Our main tool is a detailed analysis of Carleson's construction in [C]. The first author gratefully acknowledges support by the LMS and Proyecto BFM2002-04013. The second author gratefully acknowledges support by EPSRC and by the Nuffield Foundation.     

A condition for the existence of ovals in PG(2, q), q even

A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q–4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.