×

General Galois geometries. (English) Zbl 0789.51001

Oxford Mathematical Monographs. Oxford: Clarendon Press. xii, 407 p. (1991).
The book under review is the third and last volume of a treatise on projective spaces over a finite field. This trilogy is the most complete work to date on this subject, and is an indispensable reference for anyone working in the area. The first book in the series [the first author, ‘Projective geometries over finite fields’, New York: Oxford University Press (1979; Zbl 0418.51002)] dealt mainly with projective planes over the finite field \(\mathrm{GF}(q)\), although some general introductory material was also presented. The second book in the series [the first author, ‘Finite projective spaces of three dimensions’, Oxford Univ. Press, New York (1985; Zbl 0574.51001)] dealt primarily with finite projective 3- space as its title implies. The present book deals with \(\mathrm{PG}(n,q)\) for arbitrary dimension \(n\). In all cases the approach taken is one that might reasonably be called “finite algebraic geometry”. That is, the group theoretic point of view is not emphasized, but rather a combinatorial approach is taken to characterize various curves and collections of subspaces in finite projective space. The main proof techniques thus involve algebraic manipulations of coordinates over finite fields and various counting strategies. It should be noted that complete proofs are given for almost all results in the three volumes. For a more group theoretic approach to finite geometry one is referred to the classic book by P. Dembowski, ‘Finite geometries’, Berlin etc.: Springer Verlag (1968; Zbl 0159.50001), although many results in the latter reference are not proven.
The main topics discussed in this third volume are quadrics, various varieties (Hermitian, Grassmann, Veronese, Segre), polar spaces, generalized quadrangles, partial geometries, arcs and caps. The chapter on quadrics extends some work done in the previous two volumes, where the properties of quadrics in two, three and five dimensions were carefully developed. The chapters characterizing Hermitian and Grassmannian varieties over finite fields are quite different than what one would see in the classical setting, where, for instance, a Hermitian manifold over the complex numbers is not an algebraic variety. However, the chapter developing the properties of Veronese and Segre varieties closely follows the classical model. The chapter on polar spaces and generalized quadrangles is one of the few places where a number of results are stated without proof. The volume concludes with an appendix listing the known results for the existence of ovoids and spreads in the finite classical polar spaces.
There are a few topics in finite geometry that are intentionally omitted in this treatise. For instance, nondesarguesian planes are not at all discussed, and the interested reader is referred to a book such as by H. Lüneburg, ‘Translation planes’, Springer, Berlin (1980; Zbl 0446.51003) for an account of this important subject. Other topics of interest today that are not discussed in this treatise include flocks of quadrics in \(\mathrm{PG}(3,q)\), generalized \(n\)-gons for \(n>4\), and Buekenhout diagram geometries. Nonetheless, this is a very encompassing piece of work, and the topics covered are discussed with incredible detail. This is certainly true for the volume under review.
The compilation of the bibliography in and of itself is a tremendous accomplishment. There are over 2000 references given in the three volumes with the most recent publications cited appearing in print in 1991.

MSC:

51-02 Research exposition (monographs, survey articles) pertaining to geometry
51E26 Other finite linear geometries
51E12 Generalized quadrangles and generalized polygons in finite geometry
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
51E21 Blocking sets, ovals, \(k\)-arcs
51E14 Finite partial geometries (general), nets, partial spreads
14M15 Grassmannians, Schubert varieties, flag manifolds
51E20 Combinatorial structures in finite projective spaces
PDFBibTeX XMLCite