The Adjunction Theory of Complex Projective Varieties (De Gruyter Expositions in Mathematics Book 16)

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021) 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 κ(ΚM + (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

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urba?ski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Aix-Marseille Université, FranceKatrin Wendland, Trinity College Dublin, Dublin, Ireland Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021) An overview of developments in the past 15 years of adjunction theory, the study of the interplay between the intrinsic geometry of a projective variety and the geometry connected with some embedding of the variety into a projective space. Topics include consequences of positivity, the Hilbert schem