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Projective geometries over finite fields. 2nd ed. (English) Zbl 0899.51002
Oxford Mathematical Monographs. Oxford: Clarendon Press. xiv, 555 p. (1998).
The present book under review is much, much more than a second edition of an earlier work (1st ed. 1979; Zbl 0418.51002) which was the first volume of a three volume set by the author. (The other two volumes are: ‘Finite projective spaces of three dimensions’ (1985; Zbl 0574.51001) and ‘General Galois Geometries’ (1991; Zbl 0789.51001), which is coauthored with J. Thas.) It is an extensive rewrite of a valuable reference. Topics covered include: Finite fields (introductory chapter), Projective spaces and algebraic varieties, Subspaces, Partitions, Canonical forms for varieties and polarities, The line, The plane, Ovals, Arcs, Arcs in ovals, Cubic curves, Arcs of higher degree, Blocking sets, Small planes.
Besides correcting errors from the first edition, the author has taken the opportunity to reorganize the material and add pertinent results found after 1979. For example, the Hasse-Weil Theorem has been moved from Chapter 10 to a new section in Chapter 2, and it is followed by the Stöhr-Voloch Theorem proven in 1988. Chapter 2 now has a new section on coding theory and connections with finite geometries. Theorem 14.9.1 has been moved to the Notes and References Section of Chapter 2.
A few other changes are: Section 9.3 on uniform arcs is omitted. Some results of Chapter 10 are moved to Chapter 8; for example, B. Segre’s result that every oval in \(\text{PG}(2, q)\), \(q\) odd, is a conic becomes Theorem 8.14. In Chapter 14, the last section on the planes \(\text{PG}(2, 2^h)\), \(h= 4,5,6,7,8\), has been replaced with two sections on the desarguesian planes \(\text{PG}(2, 11)\) and \(\text{PG}(2, 13)\). Appendix II of the first edition has been replaced with a new Appendix II giving errors found in the volume: ‘General Galois Geometries’ by the author and J. Thas (loc. cit.). Finally, an extensive bibliography containing almost 3200 items occurs at the end of the text.
This text exhibits the care and thoroughness of its author. The organization and presentation are excellent. The book is a must for everyone who works in finite geometries and associated mathematical areas. It should be on the book-shelf next to ‘Finite geometries’ by P. Dembowski (1968; Zbl 0159.50001) and ‘Handbook of incidence geometry’ (1995; Zbl 0821.00012), edited by F. Buekenhout.

51A30 Desarguesian and Pappian geometries
51-02 Research exposition (monographs, survey articles) pertaining to geometry
51E20 Combinatorial structures in finite projective spaces
51E15 Finite affine and projective planes (geometric aspects)
51A45 Incidence structures embeddable into projective geometries
51A05 General theory of linear incidence geometry and projective geometries
05B25 Combinatorial aspects of finite geometries