Particularly essential for the subsequent development of algebraic geometry was his introduction of the zeta function of such a field and his formulation of the analogue of the Riemann hypothesis for zeta functions.

Author: Igor R. Shafarevich

Publisher: Springer

ISBN: 9783642579561

Category: Mathematics

Page: 270

View: 750

The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with first volume the author has revised the text and added new material. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of the first volume and is suitable for beginning graduate students.

Berlin Heidelberg New York, 1990, Zbl. 711.16001 (Zbl 655.00002) Shafarevich, I. R. [1988]: Basic Algebraic Geometry. Vol. I, II; 2nd suppl. ed. Nauka: Moscow, 1988. English transl.: Springer-Verlag, Berlin Heidelberg New York, 1994, ...

Author: I.R. Shafarevich

Publisher: Springer Science & Business Media

ISBN: 9783642609251

Category: Mathematics

Page: 264

View: 462

This two-part volume contains numerous examples and insights on various topics. The authors have taken pains to present the material rigorously and coherently. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.

2, 175–181. Michael Schneider and Alessandro Tancredi, Almost-positive vector bundles on projective surfaces, Math. Ann. 280 (1988), no. 4, 537–547. ... Igor R. Shafarevich, Basic Algebraic Geometry. 1. Varieties in Projective Space, ...

Author: R.K. Lazarsfeld

Publisher: Springer

ISBN: 9783642188107

Category: Mathematics

Page: 385

View: 216

Two volume work containing a contemporary account on "Positivity in Algebraic Geometry". Both volumes also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete". A good deal of the material has not previously appeared in book form. Volume II is more at the research level and somewhat more specialized than Volume I. Volume II contains a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. Contains many concrete examples, applications, and pointers to further developments

How can we recognize whether a given plane algebraic curve is rational? As we shall See later, this question is connected with rather subtle concepts of algebraic geometry. 2. Connections with the Theory of Fields.

Author: I.R. Shafarevich

Publisher: Springer Science & Business Media

ISBN: 9783642962004

Category: Mathematics

Page: 440

View: 588

Algebraic geometry occupied a central place in the mathematics of the last century. The deepest results of Abel, Riemann, Weierstrass, many of the most important papers of Klein and Poincare belong to this do mam. At the end of the last and the beginning of the present century the attitude towards algebraic geometry changed abruptly. Around 1910 Klein wrote: "When I was a student, Abelian functions*-as an after-effect of Jacobi's tradition-were regarded as the undIsputed summit of mathe matics, and each of us, as a matter of course, had the ambition to forge ahead in this field. And now? The young generation hardly know what Abelian functions are." (Vorlesungen tiber die Entwicklung der Mathe matik im XIX. Jahrhundert, Springer-Verlag, Berlin 1926, Seite 312). The style of thinking that was fully developed in algebraic geometry at that time was too far removed from the set-theoretical and axio matic spirit, which then determined the development of mathematics. Several decades had to lapse before the rise of the theory of topolo gical, differentiable and complex manifolds, the general theory of fields, the theory of ideals in sufficiently general rings, and only then it became possible to construct algebraic geometry on the basis of the principles of set-theoretical mathematics. Around the middle of the present century algebraic geometry had undergone to a large extent such a reshaping process. As a result, it can again lay claim to the position it once occupied in mathematics

This second volume introduces the concept of shemes, reviews some commutative algebra and introduces projective schemes. The finiteness theorem for coherent sheaves is proved, here again the techniques of homological algebra and sheaf cohomology are needed. In the last two chapters, projective curves over an arbitrary ground field are discussed, the theory of Jacobians is developed, and the existence of the Picard scheme is proved. Finally, the author gives some outlook into further developments- for instance étale cohomology- and states some fundamental theorems.

1. 2. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. Abhyankar, S.S.: Local Analytic Geometry. Academic Press, New York (1964); MR 31–173 Abraham, R., Robbin, J.: Transversal Mappings and Flows. Benjamin, New York (1967) 3.

Author: Igor R. Shafarevich

Publisher: Springer Science & Business Media

ISBN: 9783642379567

Category: Mathematics

Page: 310

View: 739

Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.'' The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann--Roch theorem for curves, including a proof from first principles. Shafarevich's book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.

Basic Algebraic Geometry. 1. Varieties in Projective Space. Springer-Verlag, Berlin, 2nd edition, 1994. Translated from the 1988 Russian edition and with notes by Miles Reid. Igor R. Shafarevich. Basic Algebraic Geometry. 2.

Author: Thomas A. Garrity

Publisher: American Mathematical Soc.

ISBN: 9780821893968

Category: Mathematics

Page: 335

View: 113

Algebraic Geometry has been at the center of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of ex

... [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] Igor R. Shafarevich, Basic algebraic geometry. 2, second ed., Springer-Verlag, Berlin, 1994, Schemes and complex manifolds, Translated from the 1988 Russian edition by ...

Author: Tommaso de Fernex

Publisher: American Mathematical Soc.

ISBN: 9781470435776

Category: Geometry, Algebraic

Page: 655

View: 122

This is Part 1 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

[ 63 ] F.H.S. Macaulay , The algebraic theory of modular systems , Cambridge University Press , London / New York , 1916 . ... [ 83 ] I.R. Shafarevich , Basic algebraic geometry . 2. Schemes and complex manifolds , Springer - Verlag ...

Part 1 comprises Chapters 1–4. Chapter 1 reviews some basic notions of category theory and of algebraic geometry. Chapter 2 introduces representable functors, Grothendieck topologies, and sheaves; these concepts are well known, ...

Author: Barbara Fantechi

Publisher: American Mathematical Soc.

ISBN: 9780821842454

Category: Mathematics

Page: 339

View: 620

Alexander Grothendieck introduced many concepts into algebraic geometry; they turned out to be astoundingly powerful and productive and truly revolutionized the subject. Grothendieck sketched his new theories in a series of talks at the Seminaire Bourbaki between 1957 and 1962 and collected his write-ups in a volume entitled ``Fondements de la Geometrie Algebrique,'' known as FGA. Much of FGA is now common knowledge; however, some of FGA is less well known, and its full scope is familiar to few. The present book resulted from the 2003 ``Advanced School in Basic Algebraic Geometry'' at the ICTP in Trieste, Italy. The book aims to fill in Grothendieck's brief sketches. There are four themes: descent theory, Hilbert and Quot schemes, the formal existence theorem, and the Picard scheme. Most results are proved in full detail; furthermore, newer ideas are introduced to promote understanding, and many connections are drawn to newer developments. The main prerequisite is a thorough acquaintance with basic scheme theory. Thus this book is a valuable resource for anyone doing algebraic geometry.