Algebraic Curves An Introduction to Algebraic Geometry

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Free Download Algebraic Curves: An Introduction to Algebraic Geometry By William Fulton
2008 | 132 Pages | ISBN: 0805330828 | PDF | 1 MB
PrefaceThird Preface, 2008This text has been out of print for several years, with the author holding copyrights.Since I continue to hear from young algebraic geometers who used this astheir first text, I am glad now to make this edition available without charge to anyoneinterested. I am most grateful to Kwankyu Lee for making a careful LaTeX version,which was the basis of this edition; thanks also to Eugene Eisenstein for help withthe graphics.As in 1989, I have managed to resist making sweeping changes. I thank all whohave sent corrections to earlier versions, especially Grzegorz Bobi´nski for the mostrecent and thorough list. It is inevitable that this conversion has introduced somenew errors, and I and future readers will be grateful if you will send any errors youfind to me at wfulton@umich.edu.Second Preface, 1989When this book first appeared, there were few texts available to a novice in modernalgebraic geometry. Since then many introductory treatises have appeared, includingexcellent texts by Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.The past two decades have also seen a good deal of growth in our understandingof the topics covered in this text: linear series on curves, intersection theory, andthe Riemann-Roch problem. It has been tempting to rewrite the book to reflect thisprogress, but it does not seem possible to do so without abandoning its elementarycharacter and destroying its original purpose: to introduce students with a little algebrabackground to a few of the ideas of algebraic geometry and to help them gainsome appreciation both for algebraic geometry and for origins and applications ofmany of the notions of commutative algebra. If working through the book and itsexercises helps prepare a reader for any of the texts mentioned above, that will be anadded benefit.PREFACEFirst Preface, 1969Although algebraic geometry is a highly developed and thriving field of mathematics,it is notoriously difficult for the beginner to make his way into the subject.There are several texts on an undergraduate level that give an excellent treatment ofthe classical theory of plane curves, but these do not prepare the student adequatelyfor modern algebraic geometry. On the other hand, most books with a modern approachdemand considerable background in algebra and topology, often the equivalentof a year or more of graduate study. The aim of these notes is to develop thetheory of algebraic curves from the viewpoint of modern algebraic geometry, butwithout excessive prerequisites.We have assumed that the reader is familiar with some basic properties of rings,ideals, and polynomials, such as is often covered in a one-semester course in modernalgebra; additional commutative algebra is developed in later sections. Chapter1 begins with a summary of the facts we need from algebra. The rest of the chapteris concerned with basic properties of affine algebraic sets; we have given Zariski'sproof of the important Nullstellensatz.The coordinate ring, function field, and local rings of an affine variety are studiedin Chapter 2. As in any modern treatment of algebraic geometry, they play a fundamentalrole in our preparation. The general study of affine and projective varietiesis continued in Chapters 4 and 6, but only as far as necessary for our study of curves.Chapter 3 considers affine plane curves. The classical definition of the multiplicityof a point on a curve is shown to depend only on the local ring of the curve at thepoint. The intersection number of two plane curves at a point is characterized by itsproperties, and a definition in terms of a certain residue class ring of a local ring isshown to have these properties. Bézout's Theorem and Max Noether's FundamentalTheorem are the subject of Chapter 5. (Anyone familiar with the cohomology ofprojective varieties will recognize that this cohomology is implicit in our proofs.)In Chapter 7 the nonsingular model of a curve is constructed by means of blowingup points, and the correspondence between algebraic function fields on onevariable and nonsingular projective curves is established. In the concluding chapterthe algebraic approach of Chevalley is combined with the geometric reasoning ofBrill and Noether to prove the Riemann-Roch Theorem.These notes are from a course taught to Juniors at Brandeis University in 1967-68. The course was repeated (assuming all the algebra) to a group of graduate studentsduring the intensive week at the end of the Spring semester. We have retainedan essential feature of these courses by including several hundred problems. The resultsof the starred problems are used freely in the text, while the others range fromexercises to applications and extensions of the theory.From Chapter 3 on, k denotes a fixed algebraically closed field. Whenever convenient(including without comment many of the problems) we have assumed k tobe of characteristic zero. The minor adjustments necessary to extend the theory toarbitrary characteristic are discussed in an appendix.Thanks are due to Richard Weiss, a student in the course, for sharing the taskof writing the notes. He corrected many errors and improved the clarity of the text.Professor PaulMonsky provided several helpful suggestions as I taught the course."Je n'ai jamais été assez loin pour bien sentir l'application de l'algèbre à la géométrie.Je n'ai mois point cette manière d'opérer sans voir ce qu'on fait, et il me sembloit querésoudre un probleme de géométrie par les équations, c'étoit jouer un air en tournantune manivelle. La premiere fois que je trouvai par le calcul que le carré d'unbinôme étoit composé du carré de chacune de ses parties, et du double produit del'une par l'autre, malgré la justesse de ma multiplication, je n'en voulus rien croirejusqu'à ce que j'eusse fai la figure. Ce n'étoit pas que je n'eusse un grand goût pourl'algèbre en n'y considérant que la quantité abstraite; mais appliquée a l'étendue, jevoulois voir l'opération sur les lignes; autrement je n'y comprenois plus rien."Les Confessions de J.-J. Rousseau


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