Finite geometries: classical problems and recent developments - Joseph A. Thas (Rendiconti di Matematica e delle Sue Applicazioni, 2007)
In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory, cryptography and design of experiments have made the field even more attractive. Pioneering work has been done by B. Segre and each of the four topics of this paper is related to his work; two classical problems and two recent developments will be discussed. First I will mention a purely combinatorial characterization of Hermitian curves in PG(2, q^2); here, from the beginning, the considered pointset is contained in PG(2, q^2). A second approach is where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space. This will be illustrated by a characterization of the classical inversive plane in the odd case. A recent beautiful result in Galois geometry is the discovery of an infinite class of hemisystems of the Hermitian variety in PG(3, q^2), leading to new interesting classes of incidence structures, graphs and codes; before this result, just one example for GF(9), due to Segre, was known. An exemplary example of research combining combinatorics, incidence geometry, Galois geometry and group theory is the determination of embeddings of generalized polygons in finite projective spaces. As an illustration I will discuss the embedding of the generalized quadrangle of order (4,2), that is, the Hermitian variety H(3, 4), in PG(3, K) with K any commutative field.