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AuthorFischer, Gerd. author
TitleRuled Varieties [electronic resource] : An Introduction to Algebraic Differential Geometry / by Gerd Fischer, Jens Piontkowski
ImprintWiesbaden : Vieweg+Teubner Verlag, 2001
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Descript X, 142 p. online resource


Ruled varieties are unions of a family of linear spaces. They are objects of algebraic geometry as well as differential geometry, especially if the ruling is developable. This book is an introduction to both aspects, the algebraic and differential one. Starting from very elementary facts, the necessary techniques are developed, especially concerning Grassmannians and fundamental forms in a version suitable for complex projective algebraic geometry. Finally, this leads to recent results on the classification of developable ruled varieties and facts about tangent and secant varieties. Compared to many other topics of algebraic geometry, this is an area easily accessible to a graduate course


0 Review from Classical Differential and Projective Geometry -- 0.1 Developable Rulings -- 0.2 Vanishing Gauร{159} Curvature -- 0.3 Hessian Matrices -- 0.4 Classification of Developable Surfaces in ?3 -- 0.5 Developable Surfaces in ?3(?) -- 1 Grassmannians -- 1.1 Preliminaries -- 1.2 Plรผcker Coordinates -- 1.3 Incidences and Duality -- 1.4 Tangents to Grassmannians -- 1.5 Curves in Grassmannians -- 2 Ruled Varieties -- 2.1 Incidence Varieties and Duality -- 2.2 Developable Varieties -- 2.3 The Gauร{159} Map -- 2.4 The Second Fundamental Form -- 2.5 Gauร{159} Defect and Dual Defect -- 3 Tangent and Secant Varieties -- 3.1 Zakโ{128}{153}s Theorems -- 3.2 Third and Higher Fundamental Forms -- 3.3 Tangent Varieties -- 3.4 The Dimension of the Secant Variety -- List of Symbols

Mathematics Algebraic geometry Geometry Probabilities Mathematics Algebraic Geometry Probability Theory and Stochastic Processes Geometry


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