Introduction †

This book is a result of some 13 years of work on a class of algebraic surfaces which we call Zariski surfaces since Oscar Zariski and his famous student, Heisuke Hironaka, introduced us to this project at Harvard in 1970–1971. These surfaces were discovered in the 1950s by Zariski, who discusses them briefly at the end of his paper on Castelnuovo’s criterion (Zariski, 1958), and were also considered by Abhyankar, who studied them locally as important examples for the resolution problem in characteristic p > 0. Although the theory today is not and perhaps never will be definitive, we feel that the time has come to set down some of the most important currents and problems that emerge in the study of Zariski surfaces.