The aim of this survey, written by V. A. lskovskikh and Yu. G.Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic varieties 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.

The aim of this survey, written by V. A. lskovskikh and Yu. G.Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic varieties 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 Fanolskovskikh"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.

Introduction

Chapter 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

Chapter 2. Basic Properties of Fano Varieties

 2.1. Definitions, Examples and the 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

Chapter 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 and Beginning of the Proof

 3.4. Del Pezzo Varieties with Picard Number p = 1.

 Continuation of the Proof of the Main Theorem

 3.5. Del Pezzo Varieties with Picard Number p ≥ 2.

 Conclusion of the Proof of the Main Theorem

Chapter 4. Fano Threefolds with p = 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

Chapter 5. Fano Varieties of Coindex 3 with p = 1:

The Vector Bundle Method

 5.1. Fano Threefolds of Genus 6 and 8: Gushel's Approach

 5.2. A Review of Mukai's Results on the Classification of Fano Manifolds of Coindex 3

Chapter 6. Boundedness and Rational Connectedness of Fano Varieties

 6.1. Uniruledness

 6.2. Rational Connectedness of Fano Varieties

Chapter 7. Fano Varieties with p ≥ 2

 7.1. Fano Threefolds with Picard Number p ≥ 2 (Survey of Results of Mori and Mukai

 7.2. A Survey of Results about Higher-dimensional Fano Varieties with Picard Number p ≥ 2

Chapter 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

Chapter 9. Rationality Questions for Fano Varieties II

 9.1. Birational Automorphisms of Fano Varieties

 9.2. Decomposition of Birational Maps in the Context of Mori Theory

Chapter 10. Some 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

Chapter 11. Some Particular Results and Open Problems

 11.1. On the Classification of Three-dimensional  -Fano Varieties

 11.2. Generalizations

 11.3. Some Particular Results

 11.4. Some Open Problems

Chapter 12. Appendix: Tables

 12.1. Del Pezzo Manifolds

 12.2. Fano Threefolds with p = 1

 12.3. Fano Threefolds with p = 2

 12.4. Fano Threefolds with p = 3

 12.5. Fano Threefolds with p = 4

 12.6. Fano Threefolds with p ≥ 5

 12.7. Fano Fourfolds of Index 2 with p ≥ 2

 12.8. Toric Fano Threefolds

References

Index