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.

Autorentext

Prof. Dr. em. Gerd Fischer war viele Jahre Professor für Mathematik an der Universität Düsseldorf. Er ist jetzt Gastprofessor an der Fakultät für Mathematik der TU München. Gerd Fischer ist Autor zahlreicher erfolgreicher Lehrbücher, u.a. der Linearen Algebra (vieweg studium - Grundkurs Mathematik).
Dr. Jens Piontkowski ist Hochschuldozent am Mathematischen Institut der Heinrich-Heine-Universität Düsseldorf.

Inhalt

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.- 1.1.1 Algebraic Varieties.- 1.1.2 Rational Maps.- 1.1.3 Holomorphic Linear Combinations.- 1.1.4 Limit Direction of a Holomorphic Path.- 1.1.5 Radial Paths.- 1.2 Plücker Coordinates.- 1.2.1 Local Coordinates.- 1.2.2 The Plücker Embedding.- 1.2.3 Lines in ?3.- 1.2.4 The Plücker Image.- 1.2.5 Plücker Relations.- 1.2.6 Systems of Vector Valued Functions.- 1.3 Incidences and Duality.- 1.3.1 Equations and Generators in Terms of Plücker Coordinates.- 1.3.2 Flag Varieties.- 1.3.3 Duality of Grassmannians.- 1.3.4 Dual Projective Spaces.- 1.4 Tangents to Grassmannians.- 1.4.1 Tangents to Projective Space.- 1.4.2 The Tangent Space of the Grassmannian.- 1.5 Curves in Grassmannians.- 1.5.1 The Drill.- 1.5.2 Derived Curves.- 1.5.3 Sums and Intersections.- 1.5.4 Associated Curves and Curves with Prescribed Drill.- 1.5.5 Normal Form.- 2 Ruled Varieties.- 2.1 Incidence Varieties and Duality.- 2.1.1 Unions of Linear Varieties.- 2.1.2 Fano Varieties.- 2.1.3 Joins.- 2.1.4 Conormal Bundle and Dual Variety.- 2.1.5 Duality Theorem.- 2.1.6 The Contact Locus.- 2.1.7 The Dual Curve.- 2.1.8 Rational Curves.- 2.2 Developable Varieties.- 2.2.1 Rulings.- 2.2.2 Adapted Parameterizations.- 2.2.3 Germs of Rulings.- 2.2.4 Developable Rulings and Focal Points.- 2.2.5 Developability of Joins.- 2.2.6 Dual Varieties of Cones and Degenerate Varieties.- 2.2.7 Tangent and Osculating Scrolls.- 2.2.8 Classification of Developable One Parameter Rulings.- 2.2.9 Example of a "Twisted Plane".- 2.2.10 Characterization of Drill One Curves.- 2.3 The Gauß Map.- 2.3.1 Definition of the Gauß Map.- 2.3.2 Linearity of the Fibers.- 2.3.3 Gauß Map and Developability.- 2.3.4 Gauß Image and Dual Variety.- 2.3.5 Existence of Varieties with Given Gauß Rank.- 2.4 The Second Fundamental Form.- 2.4.1 Definition of the Second Fundamental Form.- 2.4.2 The Degeneracy Space.- 2.4.3 The Degeneracy Map.- 2.4.4 The Singular and Base Locus.- 2.4.5 The Codimension of a Uniruled Variety.- 2.4.6 Fibers of the Gauß Map.- 2.4.7 Characterization of Gauß Images.- 2.4.8 Singularities of the Gauß Map.- 2.5 Gauß Defect and Dual Defect.- 2.5.1 Dual Defect of Segre Varieties.- 2.5.2 Gauß Defect and Singular Locus.- 2.5.3 Dual Defect and Singular Locus.- 2.5.4 Computation of the Dual Defect.- 2.5.5 The Surface Case.- 2.5.6 Classification of Developable Hypersurfaces.- 2.5.7 Dual Defect of Uniruled Varieties.- 2.5.8 Varieties with Very Small Dual Varieties.- 3 Tangent and Secant Varieties.- 3.1 Zak's Theorems.- 3.1.1 Tangent Spaces, Tangent Cones, and Tangent Stars.- 3.1.2 Zak's Theorem on Tangent and Secant Varieties.- 3.1.3 Theorem on Tangencies.- 3.2 Third and Higher Fundamental Forms.- 3.2.1 Definition.- 3.2.2 Vanishing of Fundamental Forms.- 3.3 Tangent Varieties.- 3.3.1 The Dimension of the Tangent Variety.- 3.3.2 Developability of the Tangent Variety.- 3.3.3 Singularities of the Tangent Variety.- 3.4 The Dimension of the Secant Variety.- List of Symbols.