Algebraic Geometry: An Introduction (Universitext)
Daniel PerrinAlgebraic GeometryAn IntroductionTranslated from the French byCatriona Maclean
Professor Daniel PerrinDépartement de MathématiquesBât. 425Faculté des Sciences d’OrsayUniversité Paris-Sud 11F-91405 Orsay CedexFranceTranslator:Catriona MacleanInstitut FourierUFR de MathématiquesUMR 5582 CNRS/Université J. Fourier100 rue des Maths, B.P. 74F-38402 St Martin d’Hères CedexFranceISBN: 978-1-84800-055-1DOI: 10.1007/978-1-84800-056-8e-ISBN: 978-1-84800-056-8British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Control Number: 2007935214Mathematics Subject Classification (2000): 14-01, 14Axx, 14H50, 14M06EDP Sciences ISBN 978-2-7598-0048-3Translation from the French language edition:Géométrie algébrique by Daniel PerrincCopyright 1995EDP Sciences, CNRS Editions, ditions.fr/All Rights Reservedc Springer-Verlag London Limited 2008 Apart from any fair dealing for the purposes of research or private study, or criticism orreview, as permitted under the Copyright, Designs and Patents Act 1988, this publicationmay only be reproduced, stored or transmitted, in any form or by any means, with theprior permission in writing of the publishers, or in the case of reprographic reproduction inaccordance with the terms of licences issued by the Copyright Licensing Agency. Enquiriesconcerning reproduction outside those terms should be sent to the publishers.The use of registered names, trademarks, etc. in this publication does not imply, even inthe absence of a specific statement, that such names are exempt from the relevant laws andregulations and therefore free for general use.The publisher makes no representation, express or implied, with regard to the accuracy ofthe information contained in this book and cannot accept any legal responsibility or liabilityfor any errors or omissions that may be made.Printed on acid-free paper9 8 7 6 5 4 3 2 1Springer Science Business Mediaspringer.com
ContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IXNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Some objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Some problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1114IAffine algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Affine algebraic sets and the Zariski topology . . . . . . . . . . . . . .2Ideal of an affine algebraic set . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4The Nullstellensatz (or Hilbert’s zeros theorem) . . . . . . . . . . . .5A first step towards Bézout’s theorem . . . . . . . . . . . . . . . . . . . . .6An introduction to morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99111315192024IIProjective algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Homographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Relation between affine and projective space . . . . . . . . . . . . . . .4Projective algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Ideal of a projective algebraic set . . . . . . . . . . . . . . . . . . . . . . . . .6A graded ring associated to a projective algebraic set . . . . . . .7Appendix: graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25252527272931323334
VIContentsIIISheaves and varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1The sheaf concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2The structural sheaf of an affine algebraic set . . . . . . . . . . . . . .3Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Sheaves of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Sheaves of modules on an affine algebraic variety . . . . . . . . . . .8Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Sheaves of modules on projective algebraic varieties . . . . . . . . .10 Two important exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . .11 Examples of morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373738414344474850525659606366IVDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1The topological definition and the link with algebra . . . . . . . . .2Dimension and counting equations . . . . . . . . . . . . . . . . . . . . . . . .3Morphisms and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Annex: finite morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69696972778283VTangent spaces and singular points . . . . . . . . . . . . . . . . . . . . . . .0Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Regular local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87878891939597VIBézout’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011Intersection multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012Bézout’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111VII Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131Some homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142Čech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173Vanishing theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214The cohomology of the sheaves OPn (d) . . . . . . . . . . . . . . . . . . . . 122Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
ContentsVIIVIII Arithmetic genus of curves and the weak Riemann-Rochtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310Introduction: the Euler-Poincaré characteristic . . . . . . . . . . . . . 1311Degree and genus of projective curves, Riemann-Roch 1 . . . . . 1322Divisors on a curve and Riemann-Roch 2 . . . . . . . . . . . . . . . . . . 138Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147IXRational maps, geometric genus and rational curves . . . . . . 1490Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1491Rational maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523Normalisation: the algebraic method . . . . . . . . . . . . . . . . . . . . . . 1554Affine blow-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585Global blow-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636Appendix: review of the above proofs . . . . . . . . . . . . . . . . . . . . . 170XLiaison of space curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1730Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1731Ideals and resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1742ACM curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793Liaison of space curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194AppendicesASummary of useful results from algebra . . . . . . . . . . . . . . . . . . . 1991Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1992Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043Transcendence bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064Some algebra exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207BSchemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2090Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2091Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2102Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2103What changes when we work with schemes . . . . . . . . . . . . . . . . 2114Why working with schemes is useful . . . . . . . . . . . . . . . . . . . . . . 2125A scheme-theoretic Bertini theorem . . . . . . . . . . . . . . . . . . . . . . . 213CProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Problem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Problem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Problem III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Problem IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Problem V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
VIIIContentsProblem VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Problem VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Problem VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228Problem IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Midterm, December 1991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Exam, January 1992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Exam, June 1992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Exam, January 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Exam, June 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Exam, February 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Index of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
PrefaceThis book is built upon a basic second-year masters course given in 1991–1992, 1992–1993 and 1993–1994 at the Université Paris-Sud (Orsay). Thecourse consisted of about 50 hours of classroom time, of which three-quarterswere lectures and one-quarter examples classes. It was aimed at students whohad no previous experience with algebraic geometry. Of course, in the timeavailable, it was impossible to cover more than a small part of this field. Ichose to focus on projective algebraic geometry over an algebraically closedbase field, using algebraic methods only.The basic principles of this course were as follows:1) Start with easily formulated problems with non-trivial solutions (suchas Bézout’s theorem on intersections of plane curves and the problem ofrational curves). In 1993–1994, the chapter on rational curves was replacedby the chapter on space curves.2) Use these problems to introduce the fundamental tools of algebraic geometry: dimension, singularities, sheaves, varieties and cohomology. I chosenot to explain the scheme-theoretic method other than for finite schemes(which are needed to be able to talk about intersection multiplicities). Ashort summary is given in an appendix, in which special importance isgiven 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 whoseproof is not necessary in practise. The main theorems used are collectedin a summary of results from algebra with references. Some of them aresuggested as exercises or problems.4) Do not hesitate to quote certain algebraic geometry theorems when theproof’s absence does not alter the reader’s understanding of the result.For example, this is the case for the uniqueness of cohomology or certaintechnical points in Chapter IX. More generally, in writing this book I triedto privilege understanding of phenomena over technique.
XPreface5) Provide a certain number of exercises and problems for every subjectdiscussed. The papers of all the exams for this course are given in anappendix at the end of the book.Clearly, a book on this subject cannot pretend to be original. This workis therefore largely based on existing works, particularly the books byHartshorne [H], Fulton [F], Mumford [M] and Shafarevitch [Sh].I would like to thank Mireille Martin-Deschamps for her careful reading and her remarks. I would also like to thank all those who attended thecourse and who pointed out several errors and suggested improvements, notably Abdelkader Belkilani, Nicusor Dan, Leopoldo Kulesz, Vincent Lafforgueand Thomas Péteul.I warmly thank Catriona Maclean for her careful translation of this bookinto English.And finally, it is my pleasure to thank Claude Sabbah for having acceptedthe French edition of this book in the series Savoirs Actuels and for his helpwith the editing of the final English edition.
NotationWe denote the set of positive integers (resp. the set of integers, rational numbers, real numbers or complex numbers) by N (resp. Z, Q, R or C). Wedenote by Fq the finite field with q elements.We denote the cardinal of a set E by E . We denote the integral part ofa real number by [x]. The notation np represents the binomial coefficient: nn!. pp!(n p)!By convention, this coefficient is zero whenever n p.If f : G H is a homomorphism of abelian groups (or modules or vectorspaces), we will denote by Ker f (resp. Im f , resp. Coker f ) its kernel (resp. itsimage, resp. its cokernel). We recall that by definition Coker f H/ Im f .An exact sequence of abelian groups (or modules or vector spaces)uv0 M M M 0is given by the data of two homomorphisms u, v such thata) u is injective,b) v is surjective,c) Im u Ker v.Further definitions and notations are contained in the summary of usefulresults from algebra.In the exercises and problems, the symbol ¶ indicates a difficult question.
Introduction0 Algebraic geometryAlgebraic geometry is the study of algebraic varieties: objects which are thezero locus of a polynomial or several polynomials. One might argue thatthe discipline goes back to Descartes. Many mathematicians—such as Abel,Riemann, Poincaré, M. Noether, Severi’s Italian school, and more recentlyWeil, Zariski and Chevalley—have produced brilliant work in this area. Thefield was revolutionised in the 1950s and 60s by the work of J.-P. Serre andespecially A. Grothendieck and has since developed considerably. It is now afundamental area of study, not just for its own sake but also because of itslinks with many other areas of mathematics.1 Some objectsThere are two basic categories of algebraic varieties: affine varieties and projective varieties. The latter are more interesting but require several definitions.It it is too early to give such definitions here; we will come back to them inChapter II.To define an affine variety, we take a family of polynomials Pi k[X1 , . . . , Xn ] with coefficients in a field k. The subset V of affine space k ndefined by the equations P1 · · · Pr 0 is then an affine algebraic variety.Here are some examples.a) If the polynomials Pi are all of degree 1, we get the linear affine subspaces of k n : lines, planes and so forth.b) Take n 2, r 1 and k R, so that k 2 is a real plane and V , definedby the equation P (X, Y ) 0, is a plane “curve”. For example, if P is ofdegree 2, the curves we get are conics (such as the ellipse X 2 Y 2 1 0,the hyperbole XY 1 0, or the parabola Y X 2 0).
2IntroductionIf P is of degree 3, we say that the curve is a cubic. For example, wecould consider Y 2 X 3 0 (which is a cuspidal curve, i.e., it has a cusp),X 3 Y 3 XY 0 (which is a nodal cubic, i.e., it has an ordinary doublepoint or node) or Y 2 X(X 1)(X 1) 0 (which is a non-singular cubic,also called an elliptic curve, cf. below).Fig. 1. X 3 Y 3 XY 0Fig. 2. Y 2 X 3 0Fig. 3. Y 2 X(X 1)(X 1) 0Of course, curves of every degree exist. Let us mention the following twocurves in particular: (X 2 Y 2 )2 3X 2 Y Y 3 0 (the trefoil) and (X 2 Y 2 )3 4X 2 Y 2 0 (the quadrifoil).c) In space k 3 an equation F (X, Y, Z) 0 defines a surface.For example, if F is of degree 2, we get quadric surfaces, such asthe sphere (X 2 Y 2 Z 2 1 0) or the one-sheeted hyperboloid(X 2 Y 2 Z 2 1 0).d) Generally, two equations in k 3 define a space curve. For example,Y X 2 0 and Z X 3 0 define a space cubic (the set of points of theform (u, u2 , u3 )
Introduction 0 Algebraic geometry Algebraic geometry is the study of algebraic varieties: objects which are the zero locus of a polynomial or several polynomials. One might argue that the discipline goes back to Descartes. Many mathematicians—such as Abel, Riemann, Poincar
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