his 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.
目录
Preface Acknowledgments Using This Book PART I: EXAMPLES OF VARIETIES AND MAPS LECTURE 1 Affine and Projective Varieties LECTURE 2 Regular Functions and Maps LECTURE 3 Cones, Projections, and More About Products LECTURE 4 Families and Parameter Spaces LECTURE 5 Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz LECTURE 6 Grassmannians and Related Varieties LECTURE 7 Rational Functions and Rational Maps LECTURE 8 More Examples LECTURE 9 Determinantal Varieties LECTURE 10 Algebraic Groups PART II: ATTRIBUTES OF VARIETIES LECTURE 11 Definitions of Dimension and Elementary Examples LECTURE 12 More Dimension Computations LECTURE 13 Hilbert Polynomials LECTURE 14 Smoothness and Tangent Spaces LECTURE 15 Gauss Maps, Tangential and Dual Varieties LECTURE 16 Tangent Spaces to Grassmannians LECTURE 17 Further Topics Involving Smoothness and Tangent Spaces LECTURE 18 Degree LECTURE 19 Further Examples and Applications of Degree LECTURE 20 Singular Points and Tangent Cones LECTURE 21 Parameter Spaces and Moduli Spaces LECTURE 22 QuadricsHints for Selected Exercises References Index
