Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 2. Berlin: Springer. xiii, 470 p. DM 178.00; £77.00; FF 671.00; Lit. 196590; $ 115.00; öS 1299.40; sFr. 155.00 (1998).
Intersection theory has always played a prominent role in the entire course of development of algebraic geometry. However, despite this very fact, the re-foundation of algebraic geometry in the second half of our century had not found its adequate reflection by a complete modern treatise on this subject, at least not so until 1984. Back then, in 1984, William Fulton finally put an end to this embarrassing anachronism by publishing his brilliant monograph “Intersection theory” (first edition 1984; Zbl 0541.14005) whose aim was both to develop the foundations of the subject within the modern algebro-geometric framework and to indicate the range of its classical and more recent applications. Fulton’s modern approach to intersection theory, which he had developed in collaboration with R. MacPherson, provided a much stronger and further-reaching theory than previously available. Also, his theory proved to be more fundamental and systematic than the various older ones, in that it was built up with fewer and less specific prerequisites from algebra and local algebraic geometry.
Of this sort, William Fulton’s “Intersection theory” has immediately become the one and only modern standard text and reference book on this still rapidly developing subject, and it has undisputedly maintained this protruding position in the current literature till today.
The present second edition of this important, nearly encyclopaedic treatise on modern intersection theory shows no changes in relation to the contents of the original, leaving out of account several minor corrections supplied, over the years, by some attentive readers and colleagues. As the author points out in the preface to the second edition, no attempt has been made to work in, or to survey the many developments in intersection theory since the appearance of the first edition. As for a few hints to the more recent literature, the reader is referred to the 1996 edition, of W. Fulton’s survey brochure “Introduction to intersection theory in algebraic geometry” (CBMS 54, Providence 1984; 3rd edition 1996). With a view to some more illustrations and concrete examples of modern intersection-theoretic methods, the reader might find it rewarding to consult the recent booklet “Using intersection theory” by S. Xambó Descamps [cf. Aportaciones Mat., Textos 7 (Mexico City 1996)]. Finally, an excellent survey (with further hints to the more recent literature) on the developments in arithmetic intersection theory is provided by J.-B. Bost’s Bourbaki talk [Séminaire Bourbaki, Vol. 1990/91, Exposé 731, Astérisque 201-203, 43-88 (1991; Zbl 0780.14013)].
Altogether, the present second edition of W. Fulton’s standard text on intersection theory was overdue, after more than a decade, all the more so as its outstanding significance has not suffered the slightest loss in all those years.