Algebraic Geometry de Daniel Perrin

Algebraic Geometry: Universitext

Autor Daniel Perrin Traducere de Catriona Maclean

en Limba Engleză Paperback – 25 ian 2008

Structura și metodologia acestui volum sunt riguros ancorate în experiența pedagogică de la Université Paris-Sud, fiind conceput ca un curs de masterat pentru studenții fără experiență prealabilă în geometria algebrică. Considerăm că forța acestui text rezidă în echilibrul dintre teorie și aplicații practice; materialul este organizat astfel încât să introducă instrumentele abstracte — dimensiunea, singularitățile, fasciculele și coomologia — prin prisma unor probleme clasice cu soluții non-triviale. Descoperim aici o progresie logică ce pornește de la studiul mulțimilor algebrice afine și proiective, culminând cu demonstrația formei slabe a teoremei Riemann-Roch și studiul curbelor spațiale.

Abordarea lui Daniel Perrin este una pur algebrică, limitând pe cât posibil utilizarea algebrei comutative abstracte fără o finalitate geometrică imediată. Spre deosebire de alte lucrări ale sale care explorează lingvistica media sau comunicarea digitală, acest volum reprezintă latura sa de cercetare matematică fundamentală, oferind o claritate didactică remarcabilă. Cititorii familiarizați cu An Undergraduate Primer in Algebraic Geometry de Ciro Ciliberto vor aprecia faptul că Daniel Perrin pune un accent mai redus pe demonstrațiile exhaustive de algebră comutativă, preferând să trimită la rezultate stabilite pentru a ajunge mai rapid la esența geometrică a obiectelor studiate. De asemenea, spre deosebire de Elementary Algebraic Geometry de K. Kendig, care mizează mult pe intuiția vizuală, volumul de față impune o rigoare algebrică specifică școlii franceze, fiind ideal pentru pregătirea cercetării avansate. Cuprinsul reflectă această ambiție, trecând de la spații tangente la coomologia fasciculelor, oferind totodată un apendice esențial despre schemele finite și elementele nilpotente.

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Specificații

ISBN-13: 9781848000551
ISBN-10: 1848000553
Pagini: 276
Ilustrații: XI, 263 p.
Dimensiuni: 155 x 235 x 16 mm
Greutate: 0.42 kg
Ediția:2008
Editura: SPRINGER LONDON
Colecția Universitext
Seria Universitext

Locul publicării:London, United Kingdom

Public țintă

Research

De ce să citești această carte

Recomandăm această carte studenților la matematică și cercetătorilor care doresc o introducere eficientă în geometria algebrică proiectivă. Cititorul câștigă acces la o metodologie testată la catedră, care prioritizează rezolvarea problemelor în detrimentul formalismului excesiv. Este un instrument de lucru concis, ideal pentru un curs semestrial, oferind bazele necesare pentru a înțelege structura curbelor și suprafețelor algebrice fără a se pierde în tehnicalități de algebră comutativă.

Descriere scurtă

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.

Cuprins

Affine algebraic sets.- Projective algebraic sets.- Sheaves and varieties.- Dimension.- Tangent spaces and singular points.- Bézout's theorem.- Sheaf cohomology.- Arithmetic genus of curves and the weak Riemann-Roch theorem.- Rational maps, geometric genus and rational curves.- Liaison of space curves.

Recenzii

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)

Textul de pe ultima copertă

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject and assumes only the standard background of undergraduate algebra. It is developed from a masters course given at the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.
The book starts with easily-formulated problems with non-trivial solutions – for example, Bézout’s theorem and the problem of rational curves – and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. The treatment uses as little commutative algebra as possible by quoting without proof (or proving only in special cases) theorems whose proof is not necessary in practice, the priority being to develop an understanding of the phenomena rather than a mastery of the technique. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.

Caracteristici

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