preface birational geometry chapter xv: ideal theory of mutative rings 1. ideals in a mutative 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 grou 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 编辑手记
内容简介：
the pure of thi volume i to provide an account of the modern algebraic method available for the invetigation of the birational geometry of algebraic varietie. an account of thee method ha already been publihed by profeor andre weil in hi foundation of algebraic geometry （new york， 1946）， and when profeor zariki colloquium lecture， delivered in 1947 to the american mathematical ociety， are publihed， another full account of thi branch of geometry will be available. the excue for a third work dealing with thi ubject i that the preent volume i deigned to appeal to a different cla of reader. it i written to meet the need of thoe geometer trained in the claical method of algebraic geometry who are anou to acquire the new and powerful tool provided by modern algebra， and who alo want to ee what they mean in term of idea familiar to them. thu in thi volume we are primarily concerned with method， and not with the tatement of original reult or with a unified theory of varietie. uch a pure in writing thi volume ha had everal effect on the n of the work. in the firt ce， we have confined our attention to varietie defined over a ground field without characteritic. thi i partly becaue the geometrical ignificance of the algebraic method and reult i more eaily prehended by a claical geometer in thi cae; alo， though other have hown that modern algebraic method have enabled u to make great tride in the theory of algebraic varietie over a field of finite characteritic， many of the theorem which the claical geometer regard a fundamental have only been proved， a yet， in the retricted cae.
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