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PREFACE BIRATIONAL GEOMETRY CHAPTER XV: IDEAL THEORY OF COMMUTATIVE RINGS 1. Ideals in a commutative ring 2. Prime ideals and primary ideals 3. Remainder-class rings 4. Subrings and extension rings 5. Quotient rings 6. Modules 7. Multiplieative theory of ideals 8. Integral dependence CHAPTER XVI: THE ARITHMETIC THEORY OF VARIETIES 1. Algebraic varieties in affine space 2. Ideals and varieties in affine space 3. Simple points 4. Irreducible subvarieties of Vd 5. Normal varieties in affine space 6. Proj ectively normal varieties CHAPTER XVII: VALUATION THEORY 1. Ordered Abelian groups 2. Valuations of a field 3. Residue fields 4. Valuations of algebraic function fields 5. The centre of a valuation CHAPTER XVIII: BIRATIONAL TRANSFORMATIONS 1. Birational correspond ences 2. Birational correspond ences between normal varieties 3. Monoidal transformations 4. The reduction of singularities and the Local Uniformisation Theorem 5. Some Cremona transformations 6. The Local Uniformisation Theorem : the main CaSe 7. Valuations of dimensions and rank k 8. Resolving systems 9. The reduction of the singularities of an algebraic variety BIBLIOGRAPHICAL NOTES BIBLIOGRAPHY INDEX ±à¼­ÊÖ¼Ç