Projective Geometries over Finite Fields / Edition 2

by James Hirschfeld

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 the only comprehensive treatise on 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 new edition is a complete reworking, containing extensive revisions, particularly in the chapters on generalities, the geometry of arcs in ovals, the geometry of arcs of higher degree, and blocking sets. Part I gives a survey of finite fields and an outline of the fundamental properties of projective spaces and their automorphisms; it includes the properties of algebraic varieties and curves used throughout the book and in the companion volumes. Part II covers, in an arbitrary dimension, the properties of subspaces, 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 plane; with little reference to the generalities from Parts I and II, the author revisits fundamental properties of the plane and then describes the structure of arcs and their relation to curves. This part includes chapters on blocking sets and on small planes (those with orders up to thirteen). With a comprehensive bibliography containing over 3,000 items, this volume will prove invaluable to researchers in finite geometry, coding theory and combinatorics.

Product Details

ISBN-13:	9780198502951
Publisher:	Oxford University Press
Publication date:	03/19/1998
Series:	Oxford Mathematical Monographs
Edition description:	REV
Pages:	570
Product dimensions:	9.10(w) x 6.30(h) x 1.50(d)

About the Author

University of Sussex

Table of Contents

I. Introduction1. Finite fields2. Projective spaces and algebraic varietiesII. Elementary general properties3. Subspaces4. Partitions5. Canonical forms for varieties and polaritiesIII. The line and the plane6. The line7. First properties of the plane8. Ovals9. Arithmetic of arcs of degree two10. Arcs in ovals11. Cubic curves12. Arcs of higher degree13. Blocking sets14. Small planesAppendixNotationReferences

Editorial Reviews

"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

From the Publisher