Buekenhout-Tits Unitals

A Buekenhout-Tits unital is defined to be a unital in PG(2, q²) obtained by coning the Tits ovoid using Buekenhout's parabolic method. The full linear collineation group stabilizing this unital is computed, and related design questions are also addressed. While the answers to the design questions are very similar to those obtained for Buekenhout-Metz unitals, the group theoretic results are quite different     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

Generalising a characterisation of Hermitian curves

This article proves a characterisation of the classical unital that is a generalisation of a characterisation proved in 1982 by Lefèvre-Percsy. It is shown that if is a Buekenhout-Metz unital with respect to a line in such that a line of not through meets in a Baer subline, then is classical. An immediate corollary is that if is a unital in PG such that is Buekenhout-Metz with respect to two distinct lines, then is classical. Received 5 August 1999; revised 15 February 2000.     

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.     

Subregular Spreads of Hermitian Unitals

In this paper, we consider the problem of constructing partitions of the points of a Hermitian unital into pairwise disjoint blocks, commonly known as spreads. We generalize a construction of Baker et al. (In Finite Geometry and Combinatorics, Vol. 191 of London Math. Soc. Lecture Not Ser., pages 17–30. Cambridge University Press, Cambridge, 1993.) to provide a new infinite family of spreads. Morover, we develop a structural connection between these new spreads of the Hermitian unital in PG(2, q²) and the subregular spreads of PG(3, q), allowing us to christen a new “subregular” family of spreads in the Hermitian unital in PG(2, q²).     

On the dimensions of the binary codes of a class of unitals

Let U_β be the special Buekenhout-Metz unital in PG(2,q²), formed by a union of q conics, where q=p^e is an odd prime power. It can be shown that the dimension of the binary code of the corresponding unital design U_β is less than or equal to q³+1−q. Baker and Wantz conjectured that equality holds. We prove that the aforementioned dimension is greater than or equal to .     

Computing Linear Codes and Unitals

The 2-rank of any 2-(28,4,1) design (unital on 28 points) is known to be between 19 and 27. It is shown by the enumeration and analysis of certain binary linear codes that there are no unitals of 2-rank 20, and that there are exactly 4 isomorphism classes of unitals of 2-rank 21. Combined with previous results, this completes the classification of unitals on 28 points of 2-rank less than 22.     

Characterizing the Hermitian and Ree Unitals on 28 Points

We prove a conjecture of Brouwer, namely that a 2-(28,4,1) design has 2–rank at least 19, with equality occuring if and only if the design is the Ree unital. We give a similar characterization of the Hermitian unital.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q²).     

Self-Orthogonal Compact Spreads and Unitals in Topological Translation Planes

A spread of is a set of l-dimensional subspaces L V partitioning V {0}. We construct examples of compact spreads that are identical with their sets of orthogonal spaces L . In the corresponding topological translation planes, every Euclidean sphere is a unital with the additional property that every point at infinity has flat feet.     

On the intersection of Hermitian curves and of Hermitian surfaces

Kestenband [Unital intersections in finite projective planes, Geom. Dedicata 11(1) (1981) 107–117; Degenerate unital intersections in finite projective planes, Geom. Dedicata 13(1) (1982) 101–106] determines the structure of the intersection of two Hermitian curves of PG(2,q²), degenerate or not. In this paper we give a new proof of Kestenband's results. Giuzzi [Hermitian varieties over finite field, Ph.D. Thesis, University of Sussex, 2001] determines the structure of the intersection of two non-degenerate Hermitian surfaces and of PG(3,q²) when the Hermitian pencil defined by and contains at least one degenerate Hermitian surface. We give a new proof of Giuzzi's results and we obtain some new results in the open case when all the Hermitian surfaces of the Hermitian pencil are non-degenerate.     

Curves covered by the Hermitian curve

A family of maximal curves is investigated that are all quotients of the Hermitian curve. These curves provide examples of curves with the same genus, the same automorphism group and the same Weierstrass semigroup at a generic point, but that are not isomorphic.     

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.