ISBN: 3-540-60168-6
TITLE: Rational Curves on Algebraic Varieties
AUTHOR: Kollar, Janos
TOC:

Introduction 1 
Chapter I. Hilbert Schemes and Chow Varieties 5 
I.1 Construction of the Hilbert Scheme 8 
I.2 Infinitesimal Study of the Hilbert Scheme 20 
I.3 Chow Varieties 40 
I.4 Chow Varieties in Positive Characteristic 62 
I.5 Related Moduli Problems 73 
I.6 Comparing Chow Varieties and Hilbert Schemes 78 
I.7 Appendix: Some Results from Commutative Algebra 83 
Chapter II. Curves on Varieties 90 
II.1 Deformations of Morphisms of Curves 92 
II.2 Families of Rational Curves 103 
II.3 Free Morphisms of Curves 113 
II.4 Cone of Curves 121 
II.5 Bend and Break 134 
II.6 Application: A Vanishing Theorem for H^1 148 
II.7 Smoothing of Morphisms of Curves 154 
Chapter III. The Cone Theorem and Minimal Models 159 
III.1 The Cone Theorem 160 
III.2 Minimal Models of Surfaces 167 
III.3 Del Pezzo Surfaces 171 
Chapter IV. Rationally Connected Varieties 179 
IV.1 Ruled and Uniruled Varieties 181 
IV.2 Minimizing Families of Rational Curves 192 
IV.3 Rationally Connected Varieties 198 
IV.4 Growing Chains of Rational Curves 209 
IV.5 Maximal Rationally Connected Fibrations 222 
IV.6 Rationally Connected Varieties over Nonclosed Fields 228 
Chapter V. Fano Varieties 238 
V.1 Low Degree Rational Curves on Fano Varieties 239 
V.2 Boundedness of Fano Varieties 250 
V.3 Characterizations of P^n 257 
V.4 Lines on Fano Hypersurfaces 265 
V.5 Nonrational Fano Varieties 272 
Chapter VI. Appendix 285 
VI.1 Rational Curves on Exceptional Loci 286 
VI.2 Intersection Numbers of Divisors and 
Asymptotic Riemann-Roch 292 
References 308 
Index 318 
END
