This volume gives an account of the principal methods used in developing a theory of algebraic varieties in space of n dimensions. Applications of these methods are also given to some of the more important varieties which occur in projective geometry. It was originally our intention to include an account of the arithmetic theory of varieties, and of the foundations of birational geometry, but it has turned out to be more convenient to reserve these topics for a third volume. The theory of algebraic varieties developed in this volume is therefore mainly a theory of varieties in projective space.

In writing this volume we have been faced with two problems: the difficult question of what must go in and what should be left out, and the problem of the degree of generality to be aimed at. As our objective has been to give an account of the modern algebraic methods available to geometers, we have not sought generality for its own sake. There is still enough to be done in the realm of classical geometry to give these methods all the scope that could be desired, and had it been possible to confine ourselves to the classical case of geometry over the field of complex numbers, we should have been content to do so. But in order to put the classical methods on a sound basis, using algebraic methods, it is necessary to consider geometry over more general fields than the field of complex numbers.