Algebraic Geometry V

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 $Orho =1$ 3.5. Del Pezzo Varieties with the Picard Number $Orho Ogeq 2$ 4. Fano Threefolds with $Orho =1$ 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 $Orho Oge 2$ 7.1. Fano Threefolds with Picard Number $Orho Oge 2$ 7.2. Higher-diumensional Fano Manifolds with $Orho Oge 2$ 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