Ruled Varieties

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 Gauß 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 Plücker 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 Gauß Map.- 2.4 The Second Fundamental Form.- 2.5 Gauß Defect and Dual Defect.- 3 Tangent and Secant Varieties.- 3.1 Zak s Theorems.- 3.2 Third and Higher Fundamental Forms.- 3.3 Tangent Varieties.- 3.4 The Dimension of the Secant Variety.- List of Symbols.