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Algebraic Geometry : An Introduction - Catriona Maclean

Algebraic Geometry

An Introduction

By: Daniel Perrin, Catriona Maclean (Translator)

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Published: 1st November 2007
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This book is built upon a basic second-year masters course given in 1991- 1992, 1992-1993 and 1993-1994 at the UniversitĀ“ e Paris-Sud (Orsay). The course consisted of about 50 hours of classroom time, of which three-quarters were lectures and one-quarter examples classes. It was aimed at students who had no previous experience with algebraic geometry. Of course, in the time available, it was impossible to cover more than a small part of this ?eld. I chose to focus on projective algebraic geometry over an algebraically closed base ?eld, using algebraic methods only. The basic principles of this course were as follows: 1) Start with easily formulated problems with non-trivial solutions (such as BĀ“ ezout's theorem on intersections of plane curves and the problem of rationalcurves).In1993-1994, thechapteronrationalcurveswasreplaced by the chapter on space curves. 2) Use these problems to introduce the fundamental tools of algebraic ge- etry: dimension, singularities, sheaves, varieties and cohomology. I chose not to explain the scheme-theoretic method other than for ?nite schemes (which are needed to be able to talk about intersection multiplicities). A short summary is given in an appendix, in which special importance is given to the presence of nilpotent elements. 3) Use as little commutative algebra as possible by quoting without proof (or proving only in special cases) a certain number of theorems whose proof is not necessary in practise. The main theorems used are collected in a summary of results from algebra with references. Some of them are suggested as exercises or problems.

From the reviews:

"The book under review, Algebraic Geometry, by Daniel Perrin, is an introductory text on modern algebraic geometry. It is aimed to be the text for a first basic course for graduate students. ... is very nicely written (and very nicely translated into English too). ... Perrin has included many, many remarks aimed to explain and deconstruct definitions and theorems. I believe these remarks will be very valuable to the reader in order to gain the very much needed intuition for the theory." (Alvaro Lozano-Robledo, MathDL, May, 2008)

"The book under review is the faithful translation into English of D. Perrin's popular French text `Geometrie algebrique. Une introduction.' ... will be to the greatest benefit of the wide international community of students, teachers, and beginning researchers in the field of modern algebraic geometry. Also, we would like to emphasize again that this primer is perfectly suitable for a one-semester graduate course on the subject, and for profound self-study just as well." (Werner Kleinert, Zentralblatt MATH, Vol. 1132 (10), 2008)

"This is the English translation of an outstanding textbook, originally published in French. ... Appendices contain a summary of results from commutative algebra used in this book and a short introduction to scheme theory. Anyone looking for a textbook on algebraic geometry that starts with the basics and presents a lot of material in a digestible way ... will find this volume an excellent choice." (Ch. Baxa, Monatshefte fur Mathematik, Vol. 160 (4), July, 2010)

Introductionp. 1
Affine algebraic setsp. 9
Projective algebraic setsp. 25
Sheaves and varietiesp. 37
Dimensionp. 69
Tangent spaces and singular pointsp. 87
Bezout's theoremp. 101
Sheaf cohomologyp. 113
Arithmetic genus of curves and the weak Riemann-Roch theoremp. 131
Rational maps, geometric genus and rational curvesp. 149
Liaison of space curvesp. 173
Summary of useful results from algebrap. 199
Schemesp. 209
Problemsp. 215
Referencesp. 251
Index of notationsp. 253
Indexp. 255
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9781848000551
ISBN-10: 1848000553
Series: Universitext
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 263
Published: 1st November 2007
Publisher: Springer London Ltd
Country of Publication: GB
Dimensions (cm): 23.5 x 15.5  x 1.4
Weight (kg): 0.42