The Purpose of this volume is to provide an account of the modern algebrc methods avlable for the investigation of the birational geometry of algebrc varieties. An account of these methods has already been published by Professor Andre Weil in his Foundations of Algebrc Geometry （New York， 1946）， and when Professor Zariski's Colloquium Lectures， delivered in 1947 to the American Mathematical Society， are published， another full account of thiranch of geometry will be avlable. The excuse for a third work dealing with this ject is that the present volume is designed to appeal to a different class of reader. It is written to meet the needs of those geometers trned in the classical methods of algebrc geometry who are anxious to acquire the new and powerful tools provided by modern algebra， and who also want to see what they mean in terms of ideas familiar to them. Thus in this volume we are primarily concerned with methods， and not with the statement of original results or with a unified theory of varieties. Such a purpose in writing this volume has had several effects on the n of the work. In the first ce， we have confined our attention to varieties defined over a ground field without characteristic. This is partly because the geometrical significance of the algebrc methods and results is more easily comprehended by a classical geometer in this case; also， though others have shown that modern algebrc methods have enabled us to make great strides in the theory of algebrc varieties over a field of finite characteristic， many of the theorems which the classical geometer regards as fundamental have only been proved， as yet， in the restricted case.
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