Algebraic Geometry
An Introduction to Birational Geometry of Algebraic Varieties
Samenvatting
The aim of this book is to introduce the reader to the geometric theory of algebraic varieties, in particular to the birational geometry of algebraic varieties. This volume grew out of the author's book in Japanese published in 3 volumes by Iwanami, Tokyo, in 1977. While writing this English version, the author has tried to rearrange and rewrite the original material so that even beginners can read it easily without referring to other books, such as textbooks on commutative algebra. The reader is only expected to know the definition of Noetherin rings and the statement of the Hilbert basis theorem. The new chapters 1, 2, and 10 have been expanded. In particular, the exposition of D-dimension theory, although shorter, is more complete than in the old version. However, to keep the book of manageable size, the latter parts of Chapters 6, 9, and 11 have been removed. I thank Mr. A. Sevenster for encouraging me to write this new version, and Professors K. K. Kubota in Kentucky and P. M. H. Wilson in Cam bridge for their careful and critical reading of the English manuscripts and typescripts. I held seminars based on the material in this book at The University of Tokyo, where a large number of valuable comments and suggestions were given by students Iwamiya, Kawamata, Norimatsu, Tobita, Tsushima, Maeda, Sakamoto, Tsunoda, Chou, Fujiwara, Suzuki, and Matsuda.
Specificaties
Inhoudsopgave
?.- 7.3 Cohomology Groups of Coherent Sheaves on PnR.- 7.4 Ample Sheaves.- 7.5 Projective Morphisms.- 7.6 Unscrewing Lemma and Its Applications.- 7.7 Projective Normality.- 7.8 Etale Morphisms.- 7.9 Theorems of Bertini.- 7.10 Monoidal Transformations.- 8 Intersection Theory of Divisors.- 8.1 Intersection Number of Curves on a Surface.- 8.2 Riemann-Roch Theorem on an Algebraic Surface.- 8.3 Intersection Matrix of a Divisor.- 8.4 Intersection Numbers of Invertible Sheaves.- 8.5 Nakai’s Criterion on Ample Sheaves.- 9 Curves on a Nonsingular Surface.- 9.1 Quadric Transformations.- 9.2 Local Properties of Singular Points.- 9.3 Linear Pencil Theorem.- 9.4 Dual Curves and Plucker Relations.- 9.5 Decomposition of Birational Maps.- 10 D-Dimension and Kodaira Dimension of Varieties.- 10.1 D-Dimension.- 10.2 The Asymptotic Estimate for l(mD).- 10.3 Fundamental Theorems for D-Dimension.- 10.4 D-Dimensions of a K3 Surface and an Abelian Variety.- 10.5 Kodaira Dimension.- 10.6 Types of Varieties.- 10.7 Subvarieties of an Abelian Variety.- 11 Logarithmic Kodaira Dimension of Varieties.- 11.1 Logarithmic Forms.- 11.2 Logarithmic Genera.- 11.3 Reduced Divisor as a Boundary.- 11.4 Logarithmic Ramification Formula.- 11.5 Étale Endomorphisms.- 11.6 Logarithmic Canonical Fibered Varieties’.- 11.7 Finiteness of the Group SBir(V).- 11.8 Some Applications.- References.