Unitals in Projective Planes

Abstract

The monograph under review investigates unitals in finite projective planes. It is the first time that a survey of the research literature on embedded unitals has been published. Many of the most recent results have been included, and they are presented using a unified notation (a notation index is provided at the end of the monograph). Some unpublished computer results are mentioned in the text as well. The reference section is quite comprehensive and covers more than 200 publications. The topics that are covered in the book include Hermitian curves and blocking sets in PG$(2,q^2)$. Unitals are then defined as a generalization of Hermitian curves. The Bruck-Bose representation in PG $(4,q)$ of a translation plane is discussed in detail leading to constructions of spreads in PG$(3,q)$. Buekenhout's constructions for embedded unitals are presented and it is shown that all known unitals in PG$(2,q^2)$ may be obtained by one of Buekenhout's constructions. These unitals are investigated in great detail including coordinates, automorphism groups and structure. A thorough study of all known unitals in the Hall plane is complemented by a survey on all known unitals in semifield planes, nearfield planes, Figueroa planes, and Hughes planes. Geometric characterizations of the classical unitals as well as the ovoidal Buekenhout-Metz unitals are treated in detail. This includes Thas' proof of the longstanding conjecture that a unital in PG$(2,q^2)$ with collinear feet from every point of its complement in PG$(2,q^2)$ is classical. The interesting and well written book closes with a list of 25 open problems.