Geometry of the Plane Cremona Maps : Lecture Notes in Mathematics - Maria Alberich-Carramiinana

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Geometry of the Plane Cremona Maps

By: Maria Alberich-Carramiinana, M. Alberich-Carramina

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Paperback | 1 July 2009

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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 Jonquiéres maps 8.4 Resolution into quadratic maps 8.5 Resolution into ordinary quadratic maps

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Preliminaries	p. 1
Plane Cremona maps	p. 29
Clebsch's theorems and jacobian	p. 73
Composition	p. 101
Characteristic matrices	p. 127
Total principal and special homaloidal curves	p. 177
Inverse Cremona map	p. 207
Noether's factorization theorem	p. 227
References	p. 249
Subject index	p. 253
Index of notation	p. 257
Table of Contents provided by Blackwell. All Rights Reserved.

Geometry of the Plane Cremona Maps

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