Algebraic Geometry V : Fano Varieties

A. N. Parshin

ISBN 10: 3642082602 / ISBN 13: 9783642082603

Editore: Springer Berlin Heidelberg, Springer Berlin Heidelberg, 2010

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Druck auf Anfrage Neuware - Printed after ordering - The aim of this survey, written by V.A. Iskovskikh and Yu.G. Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic vareties with an ample anticanonical divisor. Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh 'double projection' method and its modifications, the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

Codice articolo 9783642082603

Dati bibliografici

• Titolo: Algebraic Geometry V : Fano Varieties
• Editore: Springer Berlin Heidelberg, Springer Berlin Heidelberg
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 3642082602
• ISBN 13: 9783642082603
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 234 L x 156 A x 15 P (millimetri) 234 millimetri di larghezza per 156 millimetri di altezza per 15 millimetri di profondità
• Peso: 395 grammi

Descrizione articolo

Riassunto

The aim of this survey, written by V.A. Iskovskikh and Yu.G. Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic vareties with an ample anticanonical divisor. Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh "double projection" method and its modifications, the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties.

This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

Contenuti

0. Introduction 1. Preliminaries 1.1. Singularities 1.2. On Numerical Geometry of Cycles 1.3. On the Mori Minimal Model Program 1.4. Results on Minimal Models in Dimension Three 2. Basic Properties of Fano Varieties 2.1. Definitions, Examples and Simplest Properties 2.2. Some General Results 2.3. Existence of Good Divisors in the Fundamental Linear System 2.4. Base Points in the Fundamental Linear System 3. Del Pezzo Varieties and Fano Varieties of Large Index 3.1. On some Preliminary Results of Fujita 3.2. Del Pezzo Varieties. Definition and preliminary Results 3.3. Nonsingular del Pezzo Varieties. Statement of the Main Theorem 3.4. Del Pezzo Varieties with the Picard Number $Örho =1$ 3.5. Del Pezzo Varieties with the Picard Number $Örho Ögeq 2$ 4. Fano Threefolds with $Örho =1$ 4.1. Elementary Rational Maps: Preliminary Results 4.2. Families of Lines and Conics on Fano Threefolds 4.3. Elementary Rational Maps with Center along a Line 4.4. Elementary Rational Maps with Center along a Conic 4.5. Elementary Rational Maps with Center at a Point 4.6. Some other Rational Maps 5. Fano Manifolds of Coindex $3$ 5.1. Fano Threefolds of Genus $6$ and $8$: Gushels Approach 5.2. Review of Mukais Results 6. Boundedness and Rational Connectedness of Fano Manifolds 6.1. Uniruledness 6.2. Rational Connectedness of Fano Manifolds 7. Fano Manifolds with $Örho Öge 2$ 7.1. Fano Threefolds with Picard Number $Örho Öge 2$ 7.2. Higher-diumensional Fano Manifolds with $Örho Öge 2$ 8. Rationality Questions for Fano Varieties I 8.1. Intermediate Jacobian and Prym Varieties 8.2. Intermediate Jacobian: the Abel--Jacobi Map 8.3. The Brauer Group as a Birational Invariant 9. Rationality Questions for Fano Varieties II 9.1. Factorization of Birational Maps 9.2. Decomposition of Birational Maps in the Context of the Mori Theory 10. General Constructions of Rationality and Unirationality 10.1. Some Constructions of Unirationality 10.2. Unirationality of Complete Intersections 10.3. Some General Constructions of Rationality 11. Some Particular Results, Generalizations and Open Problems 11.1. On the Classification of Three-dimensional Q-Fano Varieties 11.2. Generalizations 11.3. Some Particular Results 11.4. Some Open Problems Appendix: Tables References Index