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14-02 Algebraic geometry -- Research exposition (monographs, survey articles)

14C17 Algebraic geometry -- Cycles and subschemes -- Intersection theory, characteristic classes, intersection multiplicities

14J25 Algebraic geometry -- Surfaces and higher-dimensional varieties -- Special surfaces En-ligne: Springerlink | Zentralblatt | MathSciNet

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This monograph treats the geometry of configuration (or parameter) spaces of ordered complete pairs, triples and quadruples of points on a smooth complex variety. Much of the work is preparatory for its application to the enumeration of the multiple points of a map between such varieties. The monograph has two parts. The first treats pairs and triples, and then recovers the double-point formula of Ronga and Laksov and the unrefined triple-point formula of Ronga and the reviewer. Notably, the latter is proved under a more relaxed hypothesis on the genericity of the map, and the added generality is illustrated in the case of a map going from a surface to a threefold and possessing an S2-singularity. The study of triples is based on Le Barz's earlier work. The formulas are derived via lengthy explicit intersection-theoretic calculations in the spirit of work by Ran and Gaffney. The second part of the monograph is just as long as the first. It is devoted to quadruples, and presents many explicit case-by-case local analyses and many lengthy computations with coordinates and equations. This constitutes a promising first step toward a proof of the reviewer's quadruple-point formula, valid under less restrictive hypotheses; however, no version whatsoever of the formula is proved. The entire monograph is well written: the exposition is clear, the flow is smooth, and the organization is logical; the development is down-to-earth and well illustrated with figures. The subject matter will appeal to students and researchers in the areas of punctual Hilbert schemes, singularities of mappings, and enumerative algebraic geometry. (MathSciNet)

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