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Intersection theory and enumerative geometry: A decade in review. (With the collaboration of Anders Thorup on §3). (English) Zbl 0664.14031
Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 321-370 (1987).
[For the entire collection see Zbl 0626.00011.]
This well written, informative and enjoyable article treats branches of enumerative geometry where the developments have been particularly remarkable over the last 10 years. Although the emphasis is on recent results, the historical background is often presented. In many instances the understanding of the material is enhanced by showing how the techniques of today have evolved from older ideas.
This article is useful for several different groups of mathematicians from experts that want to learn the historical aspects of their field, via experts in other fields that want to get some insight into enumerative geometry, to younger persons that want a rapid introduction to the field. In addition the article will also be an excellent source for later historicians as long as they remember that the involvement and influence of the author in the field and the closeness in time, which gives the quality, completeness, enthousiasm and colour of the presentation, and which makes the article so useful, also implies a slight bias in the choice of topics and in the emphasis on specific contributions.
Without any doubt this kind of overview article is extremely valuable and it is desirable that more prominent mathematicians take the time to review their own fields in this way.
Section by section the content of the article is as follows:
§1 is a historical introduction to enumerate geometry and in particular to the topics treated in the following sections.
§2 gives a well written and very readable presentation of basic intersection theory that, among other assets, will be useful for mathematicians who want to learn the subject.
§3 is written in collaboration with Anders Thorup and gives an equally useful introduction to bivariant theory.
§4 treats ranks, duality and Plücker formulas.
§5 presents recent developments of multiple-point theory.
§6 treats applications of multiple-point formulas, like Salmon-Cayley- Noether-Zeuthen formulas for surfaces in \({\mathbb{P}}^ 3\) and the formula for the bitangents to a smooth generic surface in \({\mathbb{P}}^ 4\). It also presents the interesting recent work on the Hilbert schemes of points in \({\mathbb{P}}^ 2\) and on various kinds of secant schemes.
§7 treats the contact formula and also some work on the Galois groups of commutative problems.
§8 is devoted to a, mostly historical, review of works on quadrics and correlations.
§9 is devoted to the enumerative studies of cubics. This topics has been among the most active and interesting in enumerative geometry over the last years.
Reviewer: A.Laksov

MSC:
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14-03 History of algebraic geometry
01A60 History of mathematics in the 20th century
14N05 Projective techniques in algebraic geometry