Oscar Zariski: Collected Papers: Holomorphic Functions and Linear Systems

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.