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 Vieweg Advanced Lectures in Mathematics 2010; 615 pp; softcover ISBN-10: 3-8348-0676-5 ISBN-13: 978-3-8348-0676-5 List Price: US$91 Member Price: US$81.90 Order Code: VWALM/12 This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. The two example classes of Hilbert modular surfaces and determinantal varieties are used methodically to discuss the covered techniques. For the reader the further development of the theory yields a better understanding of these fascinating objects. The text is complemented by many exercises that serve to enhance comprehension, treat additional examples, or give an outlook on further results. This book, the first of two volumes, serves as an introductory volume on schemes. The second volume concerns the cohomology of schemes. Volume I requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. A publication of Vieweg+Teubner. The AMS is exclusive distributor in North America. Vieweg+Teubner Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan. Readership Graduate students and research mathematicians interested in algebraic geometry. Table of Contents Prevarieties Spectrum of a ring Schemes Fiber products Schemes over fields Local properties of schemes Quasi-coherent modules Representable functors Separated morphisms Finiteness conditions Vector bundles Affine and proper morphisms Projective morphisms Flat morphisms and dimension One-dimensional schemes Examples