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On classification of space curves. (Sur la classification des courbes gauches.) (French) Zbl 0717.14017
Centre National de la Recherche Scientifique. Astérisque, 184-185. Paris: Société Mathématique de France. 208 p. FF 155.00; $ 28.00 (1990).
This book is a compulsory and pleasant reading for anyone willing to work on curves in \({\mathbb{P}}^ 3\). It contains both expository parts (with many examples explicitly calculated) and a very important theorem.
The exposition (which gives all the background needed for the theorem) is centered on “minimal free resolutions”, “Hartshorne-Rao module”, the Hilbert scheme of space curves, liaison, “curves in the minimal shift”. The important theorem is a key step (from a certain point of view, “the” key step) in understanding (and construction) of space curves; it gives the curves in the minimal shift of each (non Cohen-Macaulay) even liaison class and an explicit way to obtain the ones in the other shifts, up to a deformation. The theorem was independently obtained slightly before (and in a more general form: for codimension 2 subvarieties of \({\mathbb{P}}^ n\), \(n\geq 3\)) by the reviewer, G. Bolondi and J. Migliore [“The Lazarsfeld-Rao problem for liaison classes of two-codimensional subschemes of \({\mathbb{P}}^ n\)”, Am. J. Math. 113, 117-128 (1991)]. But the method of proof contained in this book is much more powerful, leading easily to explicit constructions. All the works in this area give motivations also to some purely algebraic studies on the scheme of all possible module structures.
To the references listed in the book, the isolated reader should add: the reviewer and G. Bolondi, Proc. Am. Math. Soc. 108, No.1, 43-48 (1990; Zbl 0701.14028) and Manuscr. Math. 69, No.1, 1-18 (1990; Zbl 0715.14041); the authors, “Courbes gauches et modules de Rao” (preprint May 1991); G. Bolondi and J. Migliore, “Two codimensional subschemes of arithmetically Gorenstein varieties” (preprint July 1991).
Reviewer: E.Ballico

14H50 Plane and space curves
14M06 Linkage
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14N05 Projective techniques in algebraic geometry
14H10 Families, moduli of curves (algebraic)