Birational Geometry

  • Robert Friedman
Part of the Universitext book series (UTX)


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.


Line Bundle Base Point Rational Singularity Blow Down Dual Graph 
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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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