Lectures on Curves on an Algebraic Surface

by David Mumford

Overview

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.

Product Details

ISBN-13:	9780691079936
Publisher:	Princeton University Press
Publication date:	08/21/1966
Series:	Annals of Mathematics Studies , #59
Edition description:	New Edition
Pages:	212
Product dimensions:	5.90(w) x 9.00(h) x 0.80(d)

Table of Contents

• Frontmatter, pg. i
• INTRODUCTION, pg. vii
• CONTENTS, pg. xi
• LECTURE 1. RAW MATERIAL ON CURVES ON SURFACES, AND THE PROBLEMS SUGGESTED, pg. 1
• LECTURE 2. THE FUNDAMENTAL EXISTENCE PROBLEM AND TWO ANALYTIC PROOFS, pg. 7
• LECTURE 3. PRE-SCHEMES AND THEIR ASSOCIATED “FUNCTOR OF POINTS”, pg. 11
• LECTURE 4. USES OF THE FUNCTOR OF POINTS, pg. 17
• APPENDIX TO LECTURE 4. RE REPRESENTABLE FUNCTORS AND ZARISKI TANGENT SPACES, pg. 25
• LECTURE 5. Pro j AND INVERTIBLE SHEAVES, pg. 27
• APPENDIX TO LECTURE 5, pg. 35
• LECTURE 6. PROPERTIES OP MORPHISMS AND SHEAVES, pg. 37
• LECTURE 7. RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON Pn, pg. 47
• LECTURE 8. FLATTENING STRATIFICATIONS, pg. 55
• LECTURE 9. CARTIER DIVISORS, pg. 61
• LECTURE 10. FUNCTORIAL PROPERTIES OF EFFECTIVE CARTIER DIVISORS, pg. 69
• LECTURE 11. BACK TO THE CLASSICAL CASE, pg. 75
• LECTURE 12. THE OVER-ALL CLASSIFICATION OF CURVES ON SURFACES, pg. 83
• LECTURE 13. LINEAR SYSTEMS AND EXAMPLES, pg. 91
• LECTURE 14. SOME VANISHING THEOREMS, pg. 99
• LECTURE 15. UNIVERSAL FAMILIES OF CURVES, pg. 105
• LECTURE 16. THE METHOD OF CHOW SCHEMES, pg. 111
• LECTURE 17. GOOD CURVES, pg. 119
• LECTURE 18. THE INDEX THEOREM, pg. 127
• LECTURE 19. THE PICARD SCHEME : OUTLINE, pg. 133
• LECTURE 20. INDEPENDENT 0-CYCLES ON A SURFACE, pg. 139
• LECTURE 21. THE PICARD SCHEME: CONCLUSION, pg. 145
• LECTURE 22. THE CHARACTERISTIC MAP OP A FAMILY OP CURVES, pg. 151
• LECTURE 23. THE FUNDAMENTAL THEOREM VIA KODAIRA-SPENCER, pg. 157
• LECTURE 24. THE STRUCTURE OF Φ, pg. 161
• LECTURE 25. THE FUNDAMENTAL THEOREM VIA GROTHENDIECK-CARTIER, pg. 167
• LECTURE 26. RING SCHEMES; THE WITT SCHEME, pg. 171
• APPENDICES TO LECTURE 26 APPENDICES TO LECTURE 26, pg. 189
• LECTURE 27. THE FUNDAMENTAL THEOREM IN CHARACTERISTIC p, pg. 193
• WORKS REFERRED TO, pg. 199
• Backmatter, pg. 204