Algebraic Geometry: A First CourseSpringer Science & Business Media, 17 sept. 1992 - 328 pages 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. |
Table des matières
Lecture | 3 |
Regular Functions and Maps | 17 |
Projective Varieties | 20 |
Lecture 3 | 32 |
More Projections | 38 |
Parameter Spaces of Hypersurfaces | 44 |
Ideals of Projective Varieties | 50 |
A Little Algebra | 57 |
Lecture 16 | 200 |
The Resolution of the Generic Determinantal Variety | 206 |
Further Topics Involving Smoothness and Tangent Spaces | 211 |
Blowups Nash Blowups and the Resolution of Singularities | 219 |
Bezouts Theorem | 227 |
Lecture 19 | 239 |
Degrees of Some Grassmannians | 245 |
Lecture 20 | 251 |
Lecture 6 | 63 |
Lecture 7 | 72 |
Unirationality | 87 |
More General Determinantal Varieties | 111 |
Quotients | 123 |
Attributes of Varieties | 131 |
Immediate Examples | 138 |
Lecture 12 | 151 |
Contents | 156 |
Lecture 13 | 163 |
Lecture 14 | 174 |
Projective Tangent Spaces | 181 |
Multiplicity | 258 |
Resolution of Singularities for Curves | 264 |
Hilbert Varieties | 273 |
Curves of Degree 2 | 275 |
Lecture 22 | 282 |
Linear Spaces on Quadrics | 289 |
Families of Quadrics | 295 |
Pencils of Quadrics | 301 |
Hints for Selected Exercises | 308 |
314 | |
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Expressions et termes fréquents
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