Lectures on Curves on an Algebraic Surface

✍ Scribed by David Mumford

• Publisher: Princeton University Press
• Year: 1966
• Tongue: English
• Leaves: 220
• Series: Annals of Mathematics Studies; 59
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

FRONTMATTER
INTRODUCTION
CONTENTS
LECTURE 1: RAW MATERIAL ON CURVES ON SURFACES, AND THE PROBLEMS SUGGESTED
LECTURE 2: THE FUNDAMENTAL EXISTENCE PROBLEM AND TWO ANALYTIC PROOFS
LECTURE 3: PRE-SCHEMES AND THEIR ASSOCIATED "FUNCTOR OF POINTS"
LECTURE 4: USES OF THE FUNCTOR OF POINTS
LECTURE 5: Proj AND INVERTIBLE SHEAVES
LECTURE 6: PROPERTIES OF MORPHISMS AND SHEAVES
LECTURE 7: RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON P^n
LECTURE 8: FLATTENING STRATIFICATIONS
LECTURE 9: CARTIER DIVISORS
LECTURE 10: FUNCTORIAL PROPERTIES OF EFFECTIVE CARTIER DIVISORS
LECTURE 11: BACK TO THE CLASSICAL CASE
LECTURE 12: THE OVER-ALL CLASSIFICATION OF CURVES ON SURFACES
LECTURE 13: LINEAR SYSTEMS AND EXAMPLES
LECTURE 14: SOME VANISHING THEOREMS
LECTURE 15: UNIVERSAL FAMILIES OF CURVES
LECTURE 16: THE METHOD OF CHOW SCHEMES
LECTURE 17: GOOD CURVES
LECTURE 18: THE INDEX THEOREM
LECTURE 19: THE PICARD SCHEME: OUTLINE
LECTURE 20: INDEPENDENT 0-CYCLES ON A SURFACE
LECTURE 21: THE PICARD SCHEME: CONCLUSION
LECTURE 22: THE CHARACTERISTIC MAP OF A FAMILY OF SCHEMES
LECTURE 23: THE FUNDAMENTAL THEOREM VIA KODAIRA-SPENCER
LECTURE 24: THE STRUCTURE OF PHI
LECTURE 25: THE FUNDAMENTAL THOEREM VIA GROTHENDIECK-CARTIER
LECTURE 26: RING SCHEMES; THE WITT SCHEME
LECTURE 27: THE FUNDAMENTAL THEOREM IN CHARACTERISTIC p
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