Algebraic Geometry, Vol. 133
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This book is intended to introduce students to algebraic geometry; to give them a sense of the basic objects considered, the questions asked about them, and the sort of answers one can expect to obtain. It thus emplasizes the classical roots of the subject. For readers interested in simply seeing what the subject is about, this avoids the more technical details better treated with the most recent methods. For readers interested in pursuing the subject further, this book will provide a basis for understanding the developments of the last half century, which have put the subject on a radically new footing. Based on lectures given at Brown and Harvard Universities, this book retains the informal style of the lectures and stresses examples throughout; the theory is developed as needed. The first part is concerned with introducing basic varieties and constructions; it describes, for example, affine and projective varieties, regular and rational maps, and particular classes of varieties such as determinantal varieties and algebraic groups. The second part discusses attributes of varieties, including dimension, smoothness, tangent spaces and cones, degree, and parameter and moduli spaces. Booknews This textbook is an introduction to algebraic geometry that emphasizes the classical roots of the subject, avoiding the technical details better treated with the most recent methods. It provides a basis for understanding the developments of the last half century which have put the subject on a radically new footing. Based on lectures given at Brown and Harvard, the book retains an informal style and stresses examples. Annotation c. Book News, Inc., Portland, OR (booknews.com)
PrefaceAcknowledgmentsUsing This BookPt. IExamples of Varieties and MapsLecture 1Affine and Projective Varieties3Lecture 2Regular Functions and Maps17Lecture 3Cones, Projections, and More About Products32Lecture 4Families and Parameter Spaces41Lecture 5Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz48Lecture 6Grassmannians and Related Varieties63Lecture 7Rational Functions and Rational Maps72Lecture 8More Examples88Lecture 9Determinantal Varieties98Lecture 10Algebraic Groups114Pt. IIAttributes of VarietiesLecture 11Definitions of Dimension and Elementary Examples133Lecture 12More Dimension Computations151Lecture 13Hilbert Polynomials163Lecture 14Smoothness and Tangent Spaces174Lecture 15Gauss Maps, Tangential and Dual Varieties186Lecture 16Tangent Spaces to Grassmannians200Lecture 17Further Topics Involving Smoothness and Tangent Spaces211Lecture 18Degree224Lecture 19Further Examples and Applications of Degree239Lecture 20Singular Points and Tangent Cones251Lecture 21Parameter Spaces and Moduli Spaces266Lecture 22Quadrics282Hints for Selected Exercises308References314Index317
