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Finite geometries: classical problems and recent developments. (English) Zbl 1264.51003

Summary: 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 \(\mathrm{PG}(2,q^2)\); here, from the beginning, the considered pointset is contained in \(\mathrm{PG}(2,q^2)\). 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 \(\mathrm{PG}(3,q^2)\), leading to new interesting classes of incidence structures, graphs and codes; before this result, just one example for \(\mathrm{GF}(9)\), 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 \(\mathrm{H}(3,4)\), in \(\mathrm{PG}(3,K)\) with \(K\) any commutative field.
Editorial remark: This is a republication, commissioned by the Accad.Naz.Sci., of the paper Zbl 1134.51004.

MSC:

51E20 Combinatorial structures in finite projective spaces
51E12 Generalized quadrangles and generalized polygons in finite geometry
51E15 Finite affine and projective planes (geometric aspects)
51E21 Blocking sets, ovals, \(k\)-arcs
51E25 Other finite nonlinear geometries
51A45 Incidence structures embeddable into projective geometries
51A50 Polar geometry, symplectic spaces, orthogonal spaces
51B10 Möbius geometries
05B05 Combinatorial aspects of block designs
05B25 Combinatorial aspects of finite geometries

Citations:

Zbl 1134.51004
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