Oscar Zariski, one of the most eminent mathematicians of our time, climaxed a distinguished career by receiving the National Medal of Science. He has enriched mathematics, particularly in algebraic geometry and modern algebra, by numerous and fundamental papers. This volume is the first of four in which these papers are available in collected form. By introducing ideas from abstract algebra into algebraic geometry, Zariski undertook to rewrite its foundations completely, taking an approach that made no use whatsoever of topological or convergent power series methods and that made no appeal to vague geometric intuition. The most important characteristic of this approach toward algebraic geometry, and in particular toward the problem of resolution of singularities, is that it uses the available power of modern algebra as fully as possible not only as a source of techniques in each step of solving a specific problem but also in reformulating the problem at a fundamental level. Professor Hironaka writes, "By this type of fundamental approach (not to mention specific techniques he invented to overcome specific difficulties in the problem), he made it much easier for other mathematicians in later works to follow the tracks and make further progress." The present work contains 10 papers on foundations and 9 on the resolution of singularities that were first published between 1937 and 1967. In them, new methods are introduced that enabled Zariski to study algebraic geometry over arbitrary fields of coefficients. This broader outlook made it possible to solve certain classical problems using ideal theory and the theory of valuations that had long been regarded as too difficult to be handled. Among the basic problems whose solution is found in these papers are the local uniformization of all algebraic varieties, the reduction of singularities of 2- and 3-dimensional varieties, the introduction of the concept of normal variety which is now universally used, and the proof of "Zariski's Main Theorem." Oscar Zariski: Collected Papers is part of the series Mathematicians of Our Time, edited by Gian-Carlo Rota.

Collected Papers: Foundations of Algebraic Geometry and Resolution of Singularities (Mathematicians of Our Time, V. 2, Etc.)

Science & Math

Zariski, Oscar

Mathematics

Mit Pr

Oscar Zariski's earliest papers originally appeared in 1924 and have been followed by a steady accretion ever since. That at least four volumes are required to publish his collected papers is an index to his productiveness and persistence; that they have been collected at all is a tribute to their continuing importance in the field of algebraic geometry. The first two volumes of Zariski's papers were published in 1973. Volume I, Foundations of Algebraic Geometry and Resolution of Singularities, was edited by H. Hironaka and D. Mumford, and Volume II, Holomorphic Functions and Linear Systems, was edited by M. Artin and D. Mumford. The papers contained in this third volume were originally published between 1925 and 1966, but the heart of the book is a sequence of papers, topological in nature, that appeared during the period 1928-1937. Zariski writes that "the reader will find in the introduction by M. Artin and B. Mazur an illuminating discussion of these papers and of their impact on later work by other mathematicians. Their discussion includes, in particular, my papers dealing with the following three topics: (1) solvability in radicals of equations of certain plane curves; (2) the fundamental group of the residual space of plane algebraic curves; (3) the topology of the singularities of plane algebraic curves." M. Artin (of MIT) and B. Mazur (of Harvard) both studied under Zariski at Harvard and have since become his colleagues. Their Introduction to this volume provides a useful perspective on Zariski's topological work and on this area of topology in general. These volumes are included in the series Mathematicians of Our Time, under the general editorship of Gian-Carlo Rota. Other volumes in the series now published include papers by Paul Erdös, Einar Hille, Charles Loewner, Percy Alexander MacMahon, George Pólya, Hans Rademacher, Stanislaw Ulam, and Norbert Wiener.

Collected Papers: Topology of Curves and Survaces and Special Topics in the Theory of Algebraic Varieties

This is the second of four volumes that will eventually present the full corpus of Zariski's mathematical contributions. Like the first volume (subtitled Foundations of Algebraic Geometry and Resolution of Singularities and edited by H. Hironaka and D. Mumford), it is divided into two parts, each devoted to a large but circumscribed area of research activity. The first part, containing eight papers introduced by Artin, deals with the theory of formal holomorphic functions on algebraic varieties over fields of any characteristic. The primary concern, in Zariski's words, is "analytic properties of an algebraic variety V, either in the neighborhood of a point (strictly local theory) or - and this is the deeper aspect of the theory - in the neighborhood of an algebraic subvariety of V (semiglobal theory)." Mumford surveys the ten papers reprinted in the second part. These deal with linear systems and the Riemann-Roch theorem and its applications, again in arbitrary characteristic. The applications are primarily to algebraic surfaces and include minimal models and characterization of rational or ruled surfaces.

Oscar Zariski: Collected Papers, Volume 2: Holomorphic Functions and Linear Systems (Mathematicians of Our Time)

Literature & Fiction

History & Criticism

The MIT Press

Oscar Zariski, one of the most eminent mathematicians of our time, climaxed a distinguished career by receiving the National Medal of Science. He has enriched mathematics, particularly in algebraic geometry and modern algebra, by numerous and fundamental papers. This volume is the first of four in which these papers are available in collected form.By introducing ideas from abstract algebra into algebraic geometry, Zariski undertook to rewrite its foundations completely, taking an approach that made no use whatsoever of topological or convergent power series methods and that made no appeal to vague geometric intuition. The most important characteristic of this approach toward algebraic geometry, and in particular toward the problem of resolution of singularities, is that it uses the available power of modern algebra as fully as possible not only as a source of techniques in each step of solving a specific problem but also in reformulating the problem at a fundamental level. Professor Hironaka writes, "By this type of fundamental approach (not to mention specific techniques he invented to overcome specific difficulties in the problem), he made it much easier for other mathematicians in later works to follow the tracks and make further progress."The present work contains 10 papers on foundations and 9 on the resolution of singularities that were first published between 1937 and 1967. In them, new methods are introduced that enabled Zariski to study algebraic geometry over arbitrary fields of coefficients. This broader outlook made it possible to solve certain classical problems using ideal theory and the theory of valuations that had long been regarded as too difficult to be handled.Among the basic problems whose solution is found in these papers are the local uniformization of all algebraic varieties, the reduction of singularities of 2- and 3-dimensional varieties, the introduction of the concept of normal variety which is now universally used, and the proof of "Zariski's Main Theorem.""Oscar Zariski: Collected Papers" is part of the series Mathematicians of Our Time, edited by Gian-Carlo Rota.

Oscar Zariski: Collected Papers, Volume 4: Equisingularity on Algebraic Varieties (Mathematicians of Our Time)

Evolution

This book presents fundamentals from the theory of algebraic surfaces, including areas such as rational singularities of surfaces and their relation with Grothendieck duality theory, numerical criteria for contractibility of curves on an algebraic surface, and the problem of minimal models of surfaces. In fact, the classification of surfaces is the main scope of this book and the author presents the approach developed by Mumford and Bombieri. Chapters also cover the Zariski decomposition of effective divisors and graded algebras.

Algebraic surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete)

Springer, Verlag

From the Preface: "We have preferred to write a self-contained book which could be used in a basic graduate course of modern algebra. It is also with an eye to the student that we have tried to give full and detailed explanations in the proofs... We have also tried, this time with an eye to both the student and the mature mathematician, to give a many-sided treatment of our topics, not hesitating to offer several proofs of one and the same result when we thought that something might be learned, as to methods, from each of the proofs."

Commutative Algebra I (Graduate Texts in Mathematics)

Samuel, Pierre

Springer

A precise, fundamental study of commutative algebra, this largely self-contained treatment is the first in a two-volume set. Intended for advanced undergraduates and graduate students in mathematics, its prerequisites are the rudiments of set theory and linear algebra, including matrices and determinants. The opening chapter develops introductory notions concerning groups, rings, fields, polynomial rings, and vector spaces. Subsequent chapters feature an exposition of field theory and classical material concerning ideals and modules in arbitrary commutative rings, including detailed studies of direct sum decompositions. The final two chapters explore Noetherian rings and Dedekind domains. This work prepares readers for the more advanced topics of Volume II, which include valuation theory, polynomial and power series rings, and local algebra.

Commutative Algebra: Volume I (Dover Books on Mathematics)

Dover Publications

<div>The second text in this two-book&nbsp;series extends the classical material of Volume I, which focuses on field theory and the ideal theory of Noetherian rings and Dedekind domains. The connection of Volume II's material to algebraic geometry is stressed throughout the presentation, making this book a practical introduction to some basic concepts and the arithmetical foundations of algebraic geometry. The opening chapter deals with properties of places and is followed by a chapter that explores the classical properties of polynomial and power series rings and their applications to algebraic geometry. The final chapter examines the theory of local rings, which provides the algebraic basis for the local study of algebraic and analytical varieties. Several helpful Appendixes conclude the text.</div>

Commutative Algebra: Volume II (Dover Books on Mathematics)

Commutative Algebra, V1

Literary

Literary Licensing, LLC

Commutative Algebra, V2

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