This book is based on one-semester courses given at Harvard in 1984， at Brown in 1985， and at Harvard in 1988. It is intended to be， as the title suggests， a first introduction to the subject. Even so， a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century， the subject was practiced on a relatively concrete， down-to-earth level; the main objects of study were projective varieties， and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries.Ultimately， the subject was pushed beyond the limits of its foundations： by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners. 本書為英文版。
作者介紹
Joe Harris|責編:劉慧//高蓉
目錄
Preface Acknowledgments Using This Book PART Ⅰ：EXAMPLES OF VARIETIES AND MAPS LECTURE 1 Affine and Projective Varieties A Note About Our Field Affine Space andAffineVarieties Projective Space and Projective Varieties Linear Spaces Finite Sets Hypersurfaces Analytic Subvarieties and Submanifolds The Twisted Cubic Rational Normal Curves Determinantai Representation of the Rational Normal Curve Another Parametrization of the Rational Normal Curve The Family of Plane Conics A Synthetic Construction of the Rational Normal Curve 0ther Rational Curves Varieties Defined over Subfields of K A Note on Dimension，Smoothness，and Degree LECTURE 2 Regular Functions and Maps The Zariski Topology Regular Functions on an Affine Variety Projective Varieties Regular Maps The Veronese Map Determinantal ReDresentatiOn of Veronese Varieties Subvarieties of Veronese Varieties The Segre Maps Subvarieties of Segre Varieties Products of Varieties Graphs Fiber Products Combinations of Veronese and Segre Maps LECTURE 3 Cones，Projections，and More About Products Cones Quadrics Projections M0re Cones More Projections Constructible Sets LECTURE 4 Families and Parameter Spaces Families of Varieties The Universal Hyperplane The Universal Hyperplane Section Parameter Spaces of Hypersurfaces Universal Families of Hypersurfaces A Family of Lines LECTURE 5 Ideals of Varieties，Irreducible Decomposition，and the Nullstellensatz
Generating Ideals Ideals of Projective Varieties Irreducible Varieties and Irreducible Decomposition General Objects General Projections General Twisted Cubics Double Point Loci A Little Algebra Restatements and C『orollaries LECTURE 6 Grassmannians and Related Varieties Grassmannians Subvarieties of Grassmannians …… PART Ⅱ: ATTRIBUTES OF VARIETIES Hints for Selected Exercises References Index
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