Ruled varieties. An introduction to algebraic differential geometry (Q2716063)

Ruled surfaces are the subject of several mathematical branches and also of applications as for instance in computer graphics. They are defined as one-parameter families of lines. The realization of them is easy by wire models. They provide a large portion of inspiration in particular in computer graphics. By classical differential geometry one knows that surfaces of vanishing Gaussian curvature have a ruling that is developable, i.e. the tangent plane is the same for all points of the ruling line. This is equivalent to the fact that the surface can be covered by pieces of paper. The natural generalization of a ruled surface is the ruled variety, i.e. a variety of arbitrary dimension that is `swept out' by a moving linear subspace of ambient space. This is an extrinsic property, which only makes sense relative to an ambient affine or projective space. As in the classical case of surfaces there is a strong relation to differential geometry. NEWLINENEWLINENEWLINEThe aim of the present book is the study of ruled varieties and developable ruled varieties from the point of view of complex algebraic geometry. In order to detect developable rulings it is sufficient to consider a bilinear second fundamental form that is the differential of the Gauss map. This point of view has been used by \textit{Ph. Griffiths} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 355-452 (1979; Zbl 0426.14019)]. The purpose of the book under review is to make part of this paper more accessible, to give detailed and elementary proofs, and to report on recent progress. So the text may serve as an introduction to `bilinear' complex differential geometry as a useful tool in algebraic geometry. The book is divided into four chapters. NEWLINENEWLINENEWLINEChapter 0, ``Review from classical differential and projective geometry'', reports with elementary proofs some classical facts about developable surfaces in real affine and complex projective three-space, as the linearity of the fibres of the Gauß map and the classification of developable surfaces. NEWLINENEWLINENEWLINEChapter 1, ``Grassmannians'', contains a summary of the facts about Grassmannians. Following Griffiths and Harris [see the book cited above], there is a normal form representation of curves that correspond to one-parameter families of linear spaces. NEWLINENEWLINENEWLINEThis is used in chapter 2, ``Ruled varieties'', (the core of the book) for the classification of developable varieties of Gauß rank one. Moreover there is also a classification of developable varieties of Gauß rank two in codimension one. This requires the analysis of developability with the aid of the Gauß map and the associated second fundamental form, in particular the linearity of the fibres of the Gauß map. NEWLINENEWLINENEWLINEChapter 3, ``Tangent and secant varieties'', is devoted to the particular but important class of ruled varieties of its title. It reports several of Zak's results [see \textit{F. L. Zak}, ``Tangents and secants of algebraic varieties'', Translation of Mathematical Monographs. 127 (Providence, RI 1993; Zbl 0795.14018)], that brought deep progress in this classical subject. While the dimension of the tangent variety is computable from the second fundamental form this is not the case for the secant variety, where third order invariants become important. Because the knowledge of the right third or higher order invariants for this dimension is open, the chapter concludes with some open problems. The authors assume as prerequisites for the book basic concepts of affine and projective algebraic geometry. The relatively elementary methods used in the book are based on linear algebra with algebraic or holomorphic parameters.