Overview

This book is an account of the combinatorics of projective spaces over a finite field, with special emphasis on one and two dimensions. With its successor volumes, Finite projective spaces over three dimensions (1985), which is devoted to three dimensions, and General Galois geometries (1991), on a general dimension, it provides a comprehensive treatise of this area of mathematics. The area is interesting in itself, but is important for its applications to coding theory and statistics, and its use of group theory, algebraic geometry, and number theory. This edition is a complete reworking of the first edition. The chapters bear almost the same titles as the first edition, but every chapter has been changed. The most significant changes are to Chapters 2, 10, 12, 13, which respectively describe generalities, the geometry of arcs in ovals, the geometry of arcs of higher degree, and blocking sets. The book is divided into three parts. The first part comprises two chapters, the first of which is a survey of finite fields; the second outlines the fundamental properties of projective spaces and their automorphisms, as well as properties of algebraic varieties and curves, in particular, that are used in the rest of the book and the accompanying two volumes. Parts II and III are entirely self-contained; all proofs of results are given. The second part comprises Chapters 3 to 5. They cover, in an arbitrary dimension, the properties of subspaces such as their number and characterization, of partitions into both subspaces and subgeometries, and of quadrics and Hermitian varieties, as well as polarities. Part III is a detailed account of the line and the plane. In the plane, fundamental properties are first revisited without much resort to the generalities of Parts I and II. Then, the structure of arcs and their relation to curves is described; this includes arcs both of degree two and higher degrees. There are further chapters on blocking sets and on small planes, which means of orders up to thirteen. A comprehensive bibliography of more than 3000 items is provided. At the end of each chapter is a section, Notes and References, which attributes proofs, includes further comments, and lists every relevant reference from the bibliography.

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9780198502951

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Reviews

The first edition of this work appeared in 1979, and was immediately recognized as an outstanding contribution to the field of finite geometry; the present volume is a complete revision of the earlier work...The book can serve as the text for a basic course...or as a detailed reference work for all topics in spaces of dimension two...The work is an indispensable aid to all workers in finite geometry, from beginning students to advanced researchers. Short Book Reviews ...A complete reworking [of the first edition]. As before, the volume is concisely but clearly written, and contains a wealth of interesting material...the new trilogy, comprising the 1998, 1986 and 1991 volumes, looks set to be the standard reference work on projective spaces over finite fields for many years to come. Bulletin of the London Mathematical Society 'Snap it up !' Bulletin London mathematical Society

Seven years ago, James Hirschfeld completed his ambitious project of giving a self-contained and comprehensive account of projective spaces over finite fields. The three volumes comprising this work . . . are Projective geometries over finite fields (1979), Finite projective spaces of threedimensions (1986) and General Galois geometries (1991). . . . Dr Hirschfeld has now written a second edition of the first volume of the trilogy. This is, in fact, a complete reworking, taking account of many new results proved since 1979. . . . As before, the volume is concisely but clearly written . . . The author is not planning second editions of the successor volumes, and so the new trilogy, comprising the 1998, 1986 and 1991 volumes, looks set to be the standard reference work on projective spaces over finite fields for many years to come. The publishers are currently offering the three-volume set at just over half-price. Snap it up! --Bulletin of the London Mathematical Society

Seven years ago, James Hirschfeld completed his ambitious project of giving a self-contained and comprehensive account of projective spaces over finite fields. The three volumes comprising this work . . . are Projective geometries over finite fields (1979), Finite projective spaces of three dimensions (1986) and General Galois geometries (1991). . . . Dr Hirschfeld has now written a second edition of the first volume of the trilogy. This is, in fact, a complete reworking, taking account of many new results proved since 1979. . . . As before, the volume is concisely but clearly written . . . The author is not planning second editions of the successor volumes, and so the new trilogy, comprising the 1998, 1986 and 1991 volumes, looks set to be the standard reference work on projective spaces over finite fields for many years to come. The publishers are currently offering the three-volume set at just over half-price. Snap it up! --Bulletin of the London Mathematical Society<br>

<br> Seven years ago, James Hirschfeld completed his ambitious project of giving a self-contained and comprehensive account of projective spaces over finite fields. The three volumes comprising this work . . . are Projective geometries over finite fields (1979), Finite projective spaces of three dimensions (1986) and General Galois geometries (1991). . . . Dr Hirschfeld has now written a second edition of the first volume of the trilogy. This is, in fact, a complete reworking, taking account of many new results proved since 1979. . . . As before, the volume is concisely but clearly written . . . The author is not planning second editions of the successor volumes, and so the new trilogy, comprising the 1998, 1986 and 1991 volumes, looks set to be the standard reference work on projective spaces over finite fields for many years to come. The publishers are currently offering the three-volume set at just over half-price. Snap it up! --Bulletin of the London Mathematical Society<br>

Author Information

Dr J. W. P. Hirschfeld, School of Mathematical Sciences, University of Sussex, Brighton, BN1 9QH. Tel.: +44 1273 678080; fax: +44 1273 678097

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