This book presents fundamentals from the theory of algebraic surfaces, including areas such as rational singularities of surfaces and their relation with Grothendieck duality theory, numerical criteria for contractibility of curves on an algebraic surface, and the problem of minimal models of surfaces. In fact, the classification of surfaces is the main scope of this book and the author presents the approach developed by Mumford and Bombieri. Chapters also cover the Zariski decomposition of effective divisors and graded algebras.

Inhalt

• Intersection Theory and the Nakai-Moishezon Criterion * The Hodge Index Theorem and the Intersection Matrix of a Fiber * Criteria of Contractability and Rational Singularities * Properties of Rational Singularities * Noether's Formula, Picard Scheme, Albanese Variety, and Plurigenera * Existence of Minimal Models * Morphisms to a Curve, Elliptic and Quasielliptic Fibrations * Canonical Dimension of an Elliptic or Quasielliptic Fibration * The Classification Theorem According to Canonical Dimension * Surfaces with Canonical Dimension Zero (char(k)=/2,3) * Ruled Surfaces. The Noether-Tsen Criterion * Minimal Models of Ruled Surfaces * The Zariski decomposition and applications * Appendix: Further reading