Deenition and General Properties. the Theory of Adjoints X0. Introduction

In the classical case all unirational surfaces are rational. This was rst realized by Oscar Zariski ((20], p. 314). Prompted by Hironaka's suggestion, we began an investigation of the type of surfaces introduced by Zariski in that paper. The research was originally begun by Blass in 1970-71 at Harvard with the advice of Hironaka and Zariski, and then during 1974{1977 he continued under the direction of J. S. Milne and M. Hochster at the University of Michigan. The two remaining authors entered this study a bit later. A smooth algebraic surface X deened over an algebraically closed eld, k, of characteristic p > 0 is called a Zariski Surface (or ZS) if and only if there exists two elements x, y in the function eld of X, denoted k(X), that are algebraically independent over k and such that k(x; y) k(X) k(x 1=p ; y 1=p): The main results of the paper are as follows. First of all, section 3 answers a question posed by Zariski in 1970-71. He asked whether a ZS 34