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Finite projective spaces of three dimensions. (English) Zbl 0574.51001
Oxford Mathematical Monographs. Oxford: Clarendon Press. X, 316 p. £40.00 (1985).
This continuation of the author’s ”Projective geometry over finite fields”, (1979; Zbl 0418.51002) forms the second part of an exhaustive, 3 volume compilation of results on projective spaces over a finite field. This present volume is closely linked to the previous volume: the numbering of parts, chapters, sections and propositions carries over from the first volume, copious references are made to that volume, and the bibliography of this work consists of only material omitted from that work or published since 1977. An idea of the magnitude of the work can be obtained from the total number of references: 1398.
Unlike other encyclopaedic works where proofs are rarely presented, this book offers most proofs in considerable detail.
The seven chapters treat: lines, ovaloids and quadrics; spans, spreads and packings; k-caps; Hermitian surfaces; cubic surfaces with 27 lines; twisted cubics and k-arcs. In the useful appendices are found background material on semi-linear groups, errata for the first volume, and a most necessary 6 page index of notation. The printing is vastly superior to that of the previous volume.
Reviewer: C.W.L.Garner

MSC:
51A05 General theory of linear incidence geometry and projective geometries
51-02 Research exposition (monographs, survey articles) pertaining to geometry
51A30 Desarguesian and Pappian geometries
51E25 Other finite nonlinear geometries
05B25 Combinatorial aspects of finite geometries