38 research outputs found

    On Buekenhout-Metz unitals of even order

    Get PDF
    AbstractThe even order Buekenhout-Metz unitals are enumerated (up to projective equivalence) and their inherited collineation groups are computed. They are shown to be self-dual as designs, and certain related designs are also constructed

    An alternative construction of B-M and B-T unitals in Desarguesian planes

    Get PDF
    We present a new construction of non-classical unitals from a classical unital UU in PG(2,q2)PG(2,q^2). The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π\Pi of PG(2,q2)PG(2,q^2) with the following three properties: 1. points of Π\Pi are those of PG(2,q2)PG(2,q^2); 2. lines of Π\Pi are certain lines and conics of PG(2,q2)PG(2,q^2); 3. the points in UU form a non-classical B-M unital in Π\Pi. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.Comment: Keywords: unital, desarguesian plane 11 pages; ISSN: 0012-365

    On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals

    Full text link
    This paper addresses a number of problems concerning Buekenhout-Tits unitals in PG(2,q2)PG(2,q^2), where q=2e+1q = 2^{e+1} and e≥1e \geq 1. We show that all Buekenhout-Tits unitals are PGLPGL-equivalent (addressing an open problem in [S. Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in Mathematics. Springer, New York, 2008.]), explicitly describe their PΓLP\Gamma L-stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals. J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet of points no on ℓ∞\ell_\infty in at most four points. Finally, we show that feet of points not on ℓ∞\ell_\infty are not always a {0,1,2,4}\{0,1,2,4\}-set, in contrast to what happens for Buekenhout-Metz unitals [N. Abarz\'ua, R. Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom. 18.2 (2018), 229-236]

    Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2

    Get PDF
    A unital, that is a 2-(q^3 + 1, q + 1, 1) block-design, is embedded in a projective plane π of order q^2 if its points are points of π and its blocks are subsets of lines of π, the point-block incidences being the same as in π. Regarding unitals U which are isomorphic, as a block-design, to the classical unital, T. Szonyi and the authors recently proved that the natural embedding is the unique embedding of U into the Desarguesian plane of order q^2. In this paper we extend this uniqueness result to all unitals which are isomorphic, as block-designs, to orthogonal Buekenhout-Metz unitals
    corecore