Fischer / Piontkowski Ruled Varieties

An Introduction to Algebraic Differential Geometry
2001
ISBN: 978-3-322-80217-0
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 142 Seiten, eBook

Reihe: Advanced Lectures in Mathematics

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.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Fischer, Gerd

Piontkowski, Jens

Inhaltsverzeichnis

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.

Über Autor(innen)

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.