Foreword

Summary

Introduction

This volume is an integrated collection of papers working out several new directions of research on 3-folds under the unifying theme of explicit birational geometry. Section 3 summarises briefly the contents of the individual papers.

Mori theory is a conceptual framework for studying minimal models and the classification of varieties, and has been one of the main areas of progress in algebraic geometry since the 1980s. It offers new points of view and methods of attacking classical problems, both in classification and in birational geometry, and it raises many new problem areas. While birational geometry has inspired the work of many classical and modern mathematicians, such as L. Cremona, G. Fano, Hilda Hudson, Yu. I. Manin, V. A. Iskovskikh and many others, and while their results undoubtedly give us much fascinating experimental material as food for thought, we believe that it is only within Mori theory that this body of knowledge begins to acquire a coherent shape.

At the same time as providing adequate tools for the study of 3-folds, Mori theory enriches the classical world many times over with new examples and constructions. We can now, for example, work and play with hundreds of families of Fano 3-folds. From where we stand, we can see clearly that the classical geometers were only scratching at the surface, with little inkling of the gold mine awaiting discovery.