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

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2. lines of $\Pi$ are certain lines and conics of $PG(2,q^2)$; 3. the points in $U$ form a non-classical B-M unital in $\Pi$. 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

Unitals in PG(2,q2)PG(2,q^2) with a large 2-point stabiliser

Let \cU be a unital embedded in the Desarguesian projective plane \PG(2,q^2). Write $M$ for the subgroup of \PGL(3,q^2) which preserves \cU. We show that \cU is classical if and only if \cU has two distinct points $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.Comment: Revised version - clarified the case mu\neq\lambda^{q+1} - 7 page

On the geometry of full points of abstract unitals

The concept of full points of abstract unitals has been introduced by Korchm\'aros, Siciliano and Sz\H{o}nyi 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