Classification of Complex Algebraic Surfaces

The minimal model program, or Mori’s program, one of the great summits of algebraic geometry of the late 20-th century, had the ambitious aim to develop for higher dimensional varieties the tools and methods that worked so wonderfully for the birational theory algebraic surfaces. The program, developed by S. Mori, Y. Kawamata, J. Kollár, M. Reid, V. Shokurov and many others over the last decades, has been so successful that can even be applied to the case of complex algebraic surfaces.

 

The classical Castelnuovo-Enriques classification of algebraic surfaces is now seen in the new minimal model program language: Input an algebraic surface and the output is one of two items: either a surface birational to the given one but now with canonical class nef (a strong minimal model of the given surface) or a Mori fibre space, also birational to the given surface. The outputs are described in terms of Iitaka and Sarkisov theory.

 

The book under review spells out the details of this new approach to the classification of compact complex surfaces. Assuming only a basic background on algebraic geometry, the author introduces the reader to the language and methods of the minimal model program in the case of surfaces. The main results: rationality, base-point free, extremal contraction, abundance, and cone theorems are proved in this simpler case, including discussions of the various relationships between them. Several examples and detailed calculations illustrate the theory being developed, and a few exercises at the end of some chapters make the book accesible to an interested reader.

 

The book under review fits nicely among the classical approaches to the classification of algebraic surfaces, from Beauville's Complex Algebraic Surfaces to L. Bădescu's Algebraic Surfaces.  Reading this book feels like returning home, to the familiar realm of complex algebraic surfaces, but now with the eyes and philosophy of the minimal model program, and everything looks so new, albeit familiar.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx