Fano Varieties V.A.Iskovskikh and Yu.G.Prokhorov Translated from the Russian by Yu.G.Prokhorov and S.Tregub Contents Introduction 4 Chapter 1 Preliminaries 7 §1.1. Singularities 7 §1.2. On Numerical Geometry of Cycles 11 §1.3. On the Mori Minimal Model Program 13 §1.4. Results on Minimal Models in Dimension Three 17 Chapter 2 Basic Properties of Fano Varieties 23 §2.1. Definitions, Examples and the Simplest Properties 23 §2.2. Some General Results 34 §2.3. Existence of Good Divisors in the Fundamental Linear System 39 §2.4. Base Points in the Fundamental Linear System 47 Chapter 3. Del Pezzo Varieties and Fano Varieties of Large Index 50 §3.1. On Some Preliminary Results of Fujita 50 §3.2. Del Pezzo Varieties Definition and Preliminary Results 53 §3.3. Nonsingular del Pezzo Varieties Statement of the Main Theorem and Beginning of the Proof 54 §3.4. Del Pezzo Varieties with Picard Number p=1. Continuation of the Proof of the Main Theorem 57 §3.5. Del Pezzo Varieties with Picard Number p≥2. Conclusion of the Proof of the Main Theorem 62 Chapter 4 Fano Threefolds with p=1 65 §4.1. Elementary Rational Maps: Preliminary Results 65 §4.2. Families of Lines and Conics on Fano Threefolds 71 §4.3. Elementary Rational Maps with Center along a Line 76 §4.4. Elementary Rational Maps with Center along a Conic 86 §4.5. Elementary Rational Maps with Center at a Point 95 §4.6. Some Other Rational Maps 101 Chapter 5 Fano Varieties of Coindex 3 with p=1:The Vector Bundle Method 104 §5.1. Fano Threefolds of Genus 6 and 8: GusheFs Approach 104 §5.2. A Review of Mukai’s Results on the Classification of Fano Manifolds of Coindex 3 108 Chapter 6 Boundedness and Rational Connectedness of Fano Varieties 116 §6.1. Uniruledness 116 §6.2. Rational Connectedness of Fano Varieties 120 Chapter 7 Fano Varieties with p≥2 128 §7.1. Fano Threefolds with Picard Number p≥2 (Survey of Results of Mori and Mukai 128 §7.2. A Survey of Results about Higher-dimensional Fano Varieties with Picard Number p≥2 141 Chapter 8 Rationality Questions for Fano Varieties I 153 §8.1. Intermediate Jacobian and Prym Varieties 153 §8.2. Intermediate Jacobian: the Abel-Jacobi Map 162 §8.3. The Brauer Group as a Birational Invariant 166 Chapter 9 Rationality Questions for Fano Varieties II 170 §9.1. Birational Automorphisms of Fano Varieties 170 §9.2. Decomposition of Birational Maps in the Context of Mori Theory 178 Chapter 10 Some General Constructions of Rationality and Unirationality 183 §10.1. Some Constructions of Unirationality 183 §10.2. Unirationality of Complete Intersections 188 §10.3. Some General Constructions of Rationality 191 Chapter 11 Some Particular Results and Open Problems 196 §11.1. On the Classification of Three-dimensional Q-Fano Varieties 196 §11.2. Generalizations 203 §11.3. Some Particular Results 208 §11.4. Some Open Problems 212 Chapter 12 Appendix: Tables 214 §12.1. Del Pezzo Manifolds 214 §12.2. Fano Threefolds with p=1 214 §12.3. Fano Threefolds with p=2 217 §12.4. Fano Threefolds with p=3 220 §12.5. Fano Threefolds with p=4 223 §12.6. Fano Threefolds with p>5 224 §12.7. Fano Fourfolds of Index 2 with p≥2 225 §12.8. Toric Fano Threefolds 226 References 227 Index 246
京公网安备 11010102004214号