Oscar Zariski

  • Oscar Zariski: Collected Papers, Volume 2

    Oscar Zariski: Collected Papers, Volume 2

    Holomorphic Functions and Linear Systems

    Oscar Zariski, M. Artin, and D. Mumford

    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.

    • Hardcover $80.00
    • Paperback $50.00
  • Oscar Zariski: Collected Papers, Volume 1

    Foundations of Algebraic Geometry and Resolution Singularities

    Oscar Zariski, H. Hironaka, and D. Mumford

    Oscar Zariski, one of the most eminent mathematicians of our time, has recently climaxed a distinguished career by receiving the National Medal of Science and is now Professor Emeritus at Harvard University. 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 will now be available in collect form.

    By introducing ideas from abstract algebra into algebraic geometry, Zariski undertook to rewrite its foundations completely, taking an approach that made no use whatever 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 that “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 two- and three-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 a new series, Mathematicians of Our Time, edited by Gian-Carlo Rota.

    • Hardcover $75.00