Lectures on Curves on an Algebraic Surface. (AM-59), Volume 59 (Annals of Mathematics Studies)

These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint. CONTENTS INTRODUCTION 1: Raw Material on Curves on Surfaces, and the Problems Suggested 2: The Fundamental Existence Problem and Two Analytic Proofs 3: Pre-schemes and their Associated "Functor of Points" 4: Uses of the Functor of Points Appendix to Lecture 4: Re Representable Functors and Zariski Tangent Spaces 5: Proj and Invertible Sheaves 6: Properties of Morphisms and Sheaves 7: Resume of the Cohomology of Coherent Sheaves on Pn 8: Flattening Stratifications 9: Cartier Divisors 10: Functorial Properties of Effective Cartier Divisors 11: Back to the Classical Case 12: The Over-all Classification of Curves on Surfaces 13: Linear Systems and Examples 14: Some Vanishing Theorems 15: Universal Families of Curves 16: The Method of Chow Schemes 17: Good Curves 18: The Index Theorem 19: The Picard Scheme: Outline 20: Independent 0-cycles on a Surface 21: The Picard Scheme: Conclusion 22: The Characteristic Map of a Family of Curves 23: The Fundamental Theorem V ia Kodaira-Spencer 24: The Structure of φ 25: The Fundamental Theorem Via Grothendieck-Cartier 26: Ring Schemes: The Witt Scheme Appendix to Lecture 26: 27: The Fundamental Theorem in Characteristic p BIBLIOGRAPHY