The concept of full points of abstract unitals has been introduced by Korchmáros, Siciliano and Szőnyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a more detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order.
IntroductionAn abstract unital of order n is a 2-(n 3 + 1, n + 1, 1) design. We say that an abstract unital (X, B) is embedded in a projective plane Π if X consists of points of Π and each block b ∈ B has the form X ∩ ℓ for some line ℓ of Π. For results on projective embeddings of abstract unitals see [12] and the references therein.Let U = (X, B) be an abstract unital of order n and fix two blocks b 1 , b 2 . Using the terminology of [12], we say that P is a full point with respect to (b 1 , b 2 ) if P ∈ b 1 ∪ b 2 and for each Q ∈ b 1 , the block connecting P and Q intersects b 2 . In other words, there is a well defined projection π b 1 ,P,b 2 from b 1 to b 2 with center P . We denote by F U (b 1 , b 2 ) the set of full points of U w.r.t. the blocks b 1 , b 2 . Clearly,The structure of the paper is as follows. The main result of this paper is proved in Section 3. It shows that for any abstract unital of order q, which is projectively embedded in the Galois plane PG(2, q 2 ), the set of full points of two disjoint blocks are contained in a line. Moreover, the perspectivities of two disjoint blocks generate a semi-regular cyclic permutation group. In Section 4, we extend the results of [12] by giving a complete description on the structure of full points in the classical Hermitian unitals. Section 5 gives an overview of computational results about full points in abstract unitals of order 3 and 4, which belong to known classes [1,6,14,13]. For the computation we developed and used the GAP package UnitalSZ [16].
Combinatorial properties of the set of full points2.1. Bounds on the number of full points. We start with an easy observation on the number of full points of two blocks b 1 , b 2 of U. The result seems to be rather weak.2010 Mathematics Subject Classification. 51E20, 05B25. Key words and phrases. Abstract unital, projective embedding, perspectivities, full point.
