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, Vol. 2: Holomorphic Functions and Linear Systems (Mathematicians of Our Time)

Science & Math

Oscar Zariski

Mathematics

Mit Pr

An introduction to the theory of algebraic surfaces (Lecture notes in mathematics, 83)

Springer

Covers topics such as valuation theory; theory of polynomial and power series rings; and local algebra. This volume includes the algebro-geometric connections and applications of the purely algebraic material.

Commutative Algebra II

PIERRE SAMUEL

Moduli problems in algebraic geometry date back to Riemann's famous count of the $3g-3$ parameters needed to determine a curve of genus $g$. In this book, Zariski studies the moduli space of curves of the same equisingularity class. After setting up and reviewing the basic material, Zariski devotes one chapter to the topology of the moduli space, including an explicit determination of the rare cases when the space is compact. Chapter V looks at specific examples where the dimension of the generic component can be determined through rather concrete methods. Zariski's last chapter concerns the application of deformation theory to the moduli problem, including the determination of the dimension of the generic component for a particular family of curves. An appendix by Bernard Teissier reconsiders the moduli problem from the point of view of deformation theory. He gives new proofs of some of Zariski's results, as well as a natural construction of a compactification of the moduli space.

The Moduli Problem for Plane Branches (University Lecture Series, 39)

American Mathematical Society

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Books by Oscar Zariski