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Birational Geometry

  • Robert Friedman
Chapter
  • 1.3k Downloads
Part of the Universitext book series (UTX)

Abstract

In this chapter, we describe the basic properties of surface birational geometry. After reviewing the operation of blowing up a point on a surface X, and the relationship between the invariants of the blown up surface \( \tilde{X} \) of X, we prove the Castelnuovo criterion for blowing down a curve. Using the Castelnuovo criterion, we show that every birational morphism between two smooth surfaces is a composition of blowups, and discuss various notions of minimal models. At the end of the chapter, we discuss more general contractions to normal surfaces, with particular attention to rational singularities and rational double points.

Keywords

Line Bundle Base Point Rational Singularity Blow Down Dual Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Robert Friedman
    • 1
  1. 1.Department of MathematicsColumbia UniveristyNew YorkUSA

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