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Algebraic Geometry V V.A. Iskovskikh

Algebraic Geometry V By V.A. Iskovskikh

Algebraic Geometry V by V.A. Iskovskikh

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Summary

This EMS volume provides an exposition of the structure theory of Fano varieties, i.e. algebraic varieties with an ample anticanonical divisor. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

Algebraic Geometry V Summary

Algebraic Geometry V: Fano Varieties by V.A. Iskovskikh

This EMS volume provides an exposition of the structure theory of Fano varieties, i.e. algebraic varieties with an ample anticanonical divisor. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

Table of Contents

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 $OErho =1$ 3.5. Del Pezzo Varieties with the Picard Number $OErho OEgeq 2$ 4. Fano Threefolds with $OErho =1$ 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 $OErho OEge 2$ 7.1. Fano Threefolds with Picard Number $OErho OEge 2$ 7.2. Higher-diumensional Fano Manifolds with $OErho OEge 2$ 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

Additional information

NLS9783642082603
9783642082603
3642082602
Algebraic Geometry V: Fano Varieties by V.A. Iskovskikh
V.A. Iskovskikh
Encyclopaedia of Mathematical Sciences
New
Paperback
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
2010-12-01
247
N/A
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