Joe Harris

Algebraic Geometry

A First Course

• Gebundenes Buch

Produktbeschreibung

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.

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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.

Produktdetails

• Graduate Texts in Mathematics 133
• Verlag: Springer / Springer New York / Springer, Berlin
• Artikelnr. des Verlages: 978-0-387-97716-4
• 1992
• Seitenzahl: 356
• Erscheinungstermin: 17. September 1992
• Englisch
• Abmessung: 241mm x 160mm x 25mm
• Gewicht: 689g
• ISBN-13: 9780387977164
• ISBN-10: 0387977163
• Artikelnr.: 22105955

Inhaltsangabe

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.- 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 Quadrics.- Hints for Selected Exercises.- References.

Rezensionen

"This book succeeds brilliantly by concentrating on a number of core topics (the rational normal curve, Veronese and Segre maps, quadrics, projections, Grassmannians, scrolls, Fano varieties, etc.) and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship." MATHEMATICAL REVIEWS

J. Harris

Algebraic Geometry

A First Course

"This book succeeds brilliantly by concentrating on a number of core topics (the rational normal curve, Veronese and Segre maps, quadrics, projections, Grassmannians, scrolls, Fano varieties, etc.) and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship."-MATHEMATICAL REVIEWS

J. Harris Algebraic Geometry A First Course "This book succeeds brilliantly by concentrating on a number of core topics (the rational normal curve, Veronese and Segre maps, quadrics, projections, Grassmannians, scrolls, Fano varieties, etc.) and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship."-MATHEMATICAL REVIEWS