
The Adjunction Theory of Complex Projective Varieties
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"In fact, the book under review provides a systematic, comprehensive and utmost detailed account on classical and modern adjunction theory of complex projective varieties. The authors present a monograph, which incorporates all characteristic features of a self-contained textbook, of a research report that leads to the very recent achievements in the field, and of an encyclopedia which encompasses both history and present-day state of the matter. The authors have worked in the results from nearly 700 research papers (which appeared between 1897 and 1994), including more than 50 articles published by themselves (sometimes with co-authors), and they have managed to keep the text essentially self-contained and consistent. [.] This is, mathematically and methodically, a great example of maximum efficiency in the literature on algebraic geometry. [.] The material of the book is presented in encyclopedic thoroughness, indisputable rigour, and exemplary completeness. Quite undoubtedly, it will immediately become the standard text and reference book on adjunction theory in projective algebraic geometry." Zentralblatt für Mathematik
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Content
- Intro
- Preface
- List of tables
- Chapter 1. General background results
- 1.1 Some basic definitions
- 1.2 Surface singularities
- 1.3 On the singularities that arise in adjunction theory
- 1.4 Curves
- 1.5 Nefvalue results
- 1.6 Universal sections and discriminant varieties
- 1.7 Bertini theorems
- 1.8 Some examples
- Chapter 2. Consequences of positivity
- 2.1 k-ampleness and k-bigness
- 2.2 Vanishing theorems
- 2.3 The Lefschetz hyperplane section theorem
- 2.4 The Albanese mapping in the presence of rational singularities
- 2.5 The Hodge index theorem and the Kodaira lemma
- 2.6 Rossi's extension theorems
- 2.7 Theorems of Andreotti-Grauert and Griffiths
- Chapter 3. The basic varieties of adjunction theory
- 3.1 Recognizing projective spaces and quadrics
- 3.2 Pd-bundles
- 3.3 Special varieties arising in adjunction theory
- Chapter 4. The Hilbert scheme and extremal rays
- 4.1 Flatness, the Hilbert scheme, and limited families
- 4.2 Extremal rays and the cone theorem
- 4.3 Varieties with nonnef canonical bundle
- Chapter 5. Restrictions imposed by ample divisors
- 5.1 On the behavior of k-big and ample divisors under maps
- 5.2 Extending morphisms of ample divisors
- 5.3 Ample divisors with trivial pluricanonical systems
- 5.4 Varieties that can be ample divisors only on cones
- 5.5 Pd-bundles as ample divisors
- Chapter 6. Families of unbreakable rational curves
- 6.1 Examples
- 6.2 Families of unbreakable rational curves
- 6.3 The nonbreaking lemma
- 6.4 Morphisms of varieties covered by unbreakable rational curves
- 6.5 The classification of projective manifolds covered by lines
- 6.6 Some spannedness results
- Chapter 7. General adjunction theory
- 7.1 Spectral values
- 7.2 Polarized pairs (M, L) with nefvalue & dim M - l and M singular
- 7.3 The first reduction of a singular variety
- 7.4 The polarization of the first reduction
- 7.5 The second reduction in the smooth case
- 7.6 Properties of the first and the second reduction
- 7.7 The second reduction (X, D) with KX + (n - 3) D nef
- 7.8 The three dimensional case
- 7.9 Applications
- Chapter 8. Background for classical adjunction theory
- 8.1 Numerical implications of nonnegative Kodaira dimension
- 8.2 The double point formula for surfaces
- 8.3 Smooth double covers of irreducible quadric surfaces
- 8.4 Surfaces with one dimensional projection from a line
- 8.5 k-very ampleness
- 8.6 Surfaces with Castelnuovo curves as hyperplane sections
- 8.7 Polarized varieties (X, L) with sectional genus g(L) = h1(OX)
- 8.8 Spannedness of KX + (dim X)L for ample and spanned L
- 8.9 Polarized varieties (X, L) with sectional genus g(L) = 1
- 8.10 Classification of varieties up to degree 4
- Chapter 9. The adjunction mapping
- 9.1 Spannedness of adjoint bundles at singular points
- 9.2 The adjunction mapping
- Chapter 10. Classical adjunction theory of surfaces
- 10.1 When the adjunction mapping has lower dimensional image
- 10.2 Surfaces with sectional genus g(L) = 3
- 10.3 Very ampleness of the adjoint bundle
- 10.4 Very ampleness of the adjoint bundle for degree d = 9
- 10.5 Very ampleness of the adjoint bundle when h1(OS) & 0
- 10.6 Very ampleness of the adjoint bundle when h1(OS) = 0
- 10.7 Preservation of k-very ampleness under adjunction
- Chapter 11. Classical adjunction theory in dimension = 3
- 11.1 Some results on scrolls
- 11.2 The adjunction mapping with a lower dimensional image
- 11.3 Very ampleness of the adjoint bundle
- 11.4 Applications to hyperelliptic curve sections
- 11.5 Projective normality of adjoint bundles
- 11.6 Manifolds of sectional genus = 4
- 11.7 The Fano-Morin adjunction process
- Chapter 12. The second reduction in dimension three
- 12.1 Exceptional divisors of the second reduction morphism
- 12.2 The structure of the second reduction
- 12.3 The second reduction for threefolds in P5
- Chapter 13. Varieties (M, L) with ?(?M + (dim M - 2)L) = 0
- 13.1 The double point formula for threefolds
- 13.2 The linear system |KM + (n - 2)L| on the first reduction (M, L)
- 13.3 Some Chern inequalities for ample divisors
- Chapter 14. Special varieties
- 14.1 Structure results for scrolls
- 14.2 Structure results for quadric fibrations
- 14.3 Varieties with small invariants
- 14.4 Projective manifolds with positive defect
- 14.5 Hyperplane sections of curves
- Bibliography
- Index
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