466 research outputs found
Eggs in finite projective spaces and unitals in translation planes
No full textInspired by the connection between ovoids and unitals arising from the Buekenhout construction in the Andre/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of PG(4m - 1, q), m >= 1, are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order 310, which is represented in PG(20, 3) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in PG(19, 3). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane
On eggs and translation generalised quadrangles
No full textWe study eggs in . A new model for eggs is presented in which
all known examples are given. We calculate
the general form of the dual egg for eggs arising from a semifield flock.
Applying this to the egg obtained in L. Bader, G. Lunardon and I. Pinneri
\cite{BALUPI} from the Penttila-Williams ovoid \cite{PEWI},
we obtain the dual egg, which is not isomorphic to any of the previous known
examples, see \cite{BALUPI}. Furthermore we give a new proof of a conjecture
of J.A.Thas \cite{TH1} using our model, and
classify all eggs of which is equivalent
to the classification of all translation generalised
quadrangles of order (4,16)
Relative symplectic subquadrangle hemisystems of the Hermitian surface
No full textWe introduce the notion of relative subquadrangle regular system of a generalized quadrangle. A relative subquadrangle regular system of order m on a generalized quadrangle S of order (s, t) is a set R of embedded subquadrangles with a prescribed intersection prop- erty with respect to a given subquadrangle T such that every point of S \ T lies on exactly m subquadrangles of R. If m is one half of the total number of such subquadrangles on a point we call R a relative subquadrangle hemisystem with respect to T . We construct two infinite families of symplectic relative subquadrangle hemisystems of the Hermitian surface H(3,q2), q even
Some remarks on flocks
No full textNew proofs are given of the fundamental results of Bader,
Lunardon and Thas relating flocks of the quadratic
cone in pg(3,q), q odd, and BLT-sets of Q(4,q). We also
show that there is a unique BLT-set of H(3,9). The model of
Penttila for Q(4,q) q odd, is extended to
Q(2m,q) to construct partial flocks of size qm/2+m/2-1 of the
cone K in pg(2m-1,q) with vertex a point and base
Q(2m-2,q) , where q is congruent to 1 or 3 modulo 8 and m
is even. These
partial flocks are larger than the largest previously known
for m>2 . Also, the
example of O'Keefe and Thas of a partial
flock of K in pg(5,3) of size 6 is generalised to a
partial flock of the cone K of pg(2pn-1,p) of size 2pn , for any prime p congruent to 1 or 3 modulo 8,
with the corresponding partial BLT-set of
Q(2pn,p) admitting the symmetric group of degree 2pn+1
Subquadrangle m-regular systems on generalized quadrangles
No full textWe introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of R. If m is one half of the total number of subquadrangles on a point, we call R a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface âï(3, q2), q odd, and two infinite families of symplectic subquadrangle hemisystems of 3(q2), q even. Some sporadic examples of symplectic subquadrangle regular systems of âï H(3, q2) are also presented
Segre's hemisystem and McLaughlin's graph
No full textAbstractAn alternative construction of Segre's hemisystem of H(3,9) is provided, as well as an alternative construction of McLauhglin's strongly regular graph srg(275,112,30,56) in terms of Segre hemisystems
Hemisystems on the Hermitian surface
No full textThe natural geometric setting of quadrics commuting with a Hermitian surface of PG(3,q2), q odd, is adopted and a hemisystem on the Hermitian surface H(3,q2) admitting the group PΩ−(4,q) is constructed, yielding a partial quadrangle PQ((q−1)/2, q2,(q−1)2/2) and a strongly regular graph srg((q3+1)(q+1)/2,(q2+1)(q−1)/2,(q−3)/2,(q−1)2/2). For q>3, no partial quadrangle or strongly regular graph with these parameters was previously known, whereas when q=3, this is the Gewirtz graph. Thas conjectured that there are no hemisystems on H(3,q2) for q>3, so these are counterexamples to his conjecture. Furthermore, a hemisystem on H(3,25) admitting 3.A7.2 is constructed. Finally, special sets (after Shult) and ovoids on H(3,q2) are investigated
The action of the group G2(q) < PSU6(q2), q even, and related combinatorial structures
No full textWe describe some geometrical properties of the action of the Cartan-Dickson-Chevalley exceptional group G2(q), q even, as a subgroup of the unitary group PSU6(q^2).This allows us to provide a new description of the known 126-hyperoval of H(5,4) and to construct even subsets of PG(2,q ^2)
Two-character sets arising from bluings of orbits
No full textIn this paper we construct two-character sets as gluings of orbits of the group stabilizing a twisted cubi
Variations on a Theme of Glauberman
No full textA new and elementary proof of the Artin–Zorn theorem that finite alternative division rings are fields is given. The characterisation of finite fields of Glauberman and Heimbeck is also extended to a broader class of fields, the two subjects being connected via geometry
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