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Algebraic Geometry

An Introduction

  • Textbook
  • © 2008

Overview

Authors:
  1. Daniel Perrin

    1. Département de Mathématiques, Bât. 425 Faculté des Sciences d'Orsay Université Paris-Sud 11, Orsay Cedex, France

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  • Introduces the fundamental tools of algebraic geometry at a level suitable for beginning researchers in the domain

Part of the book series: Universitext (UTX)

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XX
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  2. Affine algebraic sets

    Pages 9-24

  3. Projective algebraic sets

    Pages 25-36

  4. Sheaves and varieties

    Pages 37-68

  5. Dimension

    Pages 69-86

  6. Tangent spaces and singular points

    Pages 87-99

  7. Bézout's theorem

    Pages 101-112

  8. Sheaf cohomology

    Pages 113-129

  9. Arithmetic genus of curves and the weak Riemann-Roch theorem

    Pages 131-148

  10. Rational maps, geometric genus and rational curves

    Pages 149-171

  11. Liaison of space curves

    Pages 173-195

  12. Back Matter

    Pages 199-263
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Keywords

About this book

This book is built upon a basic second-year masters course given in 1991– 1992, 1992–1993 and 1993–1994 at the Universit´ 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 B´ 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.

Reviews

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." (Álvaro Lozano-Robledo, MathDL, May, 2008)

"The book under review is the faithful translation into English of D. Perrin’s popular French text ‘Géométrie algébrique. 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 für Mathematik, Vol. 160 (4), July, 2010)

Authors and Affiliations

  • Département de Mathématiques, Bât. 425 Faculté des Sciences d'Orsay Université Paris-Sud 11, Orsay Cedex, France

    Daniel Perrin

Bibliographic Information

  • Book Title: Algebraic Geometry

  • Book Subtitle: An Introduction

  • Authors: Daniel Perrin

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-84800-056-8

  • Publisher: Springer London

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag London 2008

  • Softcover ISBN: 978-1-84800-055-1Published: 25 January 2008

  • eBook ISBN: 978-1-84800-056-8Published: 16 December 2007

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XI, 263

  • Additional Information: Original French edition published by EDP Sciences, CNRS Editions, France, 1995

  • Topics: Algebraic Geometry, General Algebraic Systems, Mathematics, general

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