One of the major discoveries of the last two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization is called the minimal model program or Mori's program. While originally the program was conceived with the sole aim of constructing higher dimensional analogues of minimal models of surfaces, by now it has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond.

So far the program is complete only in dimension 3, but large parts are known to work in all dimensions.

The aim of this book is to introduce the reader to the circle of ideas developed around the minimal model program, relying only on knowledge of basic algebraic geometry.

In order to achieve this goal, considerable effort was devoted to make the book as self-contained as possible. We managed to simplify many of the proofs, but in some cases a compromise seemed a better alternative. There are quite a few cases where a theorem which is local in nature is much easier to prove for projective varieties. For these, we state the general theorem and then prove the projective version, giving references for the general cases. Most of the applications of the minimal model program ultimately concern projective varieties, and for these the proofs in this book are complete

Acknowledgments

The present form of this book owes a lot to the contributions of our two collaborators.

H. Clemens was our coauthor in [CKM88]. Sections 1.1–3, 2.1, 2.2, 2.4 and 3.1–5 are revised versions of sections of [CKM88].