Geometry of the plane cremona maps
Author(s)
Bibliographic Information
Geometry of the plane cremona maps
（Lecture notes in mathematics, 1769）
Springer, c2002
Available at / 75 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
L/N||LNM||176978800463
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
DC21:516.35/AL142070552191
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Note
Bibliography: p. [249]-251
Includes index
Description and Table of Contents
Description
This book provides a self-contained exposition of the theory of plane Cremona maps, reviewing the classical theory. The book updates, correctly proves and generalises a number of classical results by allowing any configuration of singularities for the base points of the plane Cremona maps. It also presents some material which has only appeared in research papers and includes new, previously unpublished results. This book will be useful as a reference text for any researcher who is interested in the topic of plane birational maps.
Table of Contents
1. Preliminaries 1.1 Blowing-ups 1.2 Weighted clusters 1.3 Birational maps of surfaces 2. Plane Cremona maps 2.1 Base points 2.2 Principal curves 2.3 Contractile curves 2.4 Characteristic matrix 2.5 Equations of condition 2.6 Noether's inequality 2.7 Further relations 2.8 Quadratic plane Cremona maps 2.9 Transforming curves 3. Clebsch's theorems and jacobian 3.1 A Clebsch's theorem 3.2 The entries of the characteristic matrix 3.3 On symmetry of chararcteristics 3.4 Further properties 3.5 Jacobian of the homaloidal net 4. Composition 4.1 Composition of two plane Cremona maps 4.2 Consequences 5. Characteristic matrices 5.1 Homaloidal nets 5.2 Homaloidal types 5.3 On proper homaloidal types 5.4 Characteristic matrices 5.5 Exceptional types 5.6 On proper exceptional types 5.7 Weyl groups 6. Total principal and special homaloidal curves 6.1 Virtual versus effective behaviour 6.2 Non-expansive corresponding base points 6.3 Generic versus effective behaviour 6.4 Irreducible homaloidal curves 6.5 Special homaloidal curves 7 Inverse Cremona map 7.1 Non-expected contractile curves 7.2 Proximity among base points of the inverse 7.3 Inverse map and total principle curves 7.4 Consequences 8. Noether's factorization theorem 8.1 Criterion for homaloidal nets 8.2 Complexity and major base points 8.3 Resolution into the Jonquieres maps 8.4 Resolution into quadratic maps 8.5 Resolution into ordinary quadratic maps
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