Sphere Packings, Lattices and Groups

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

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Integral closure of ideals, rings, and modules

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

In this chapter, we give the definition of Mordell–Weil lattice (in Sect. 6.5). First, we bring together the concepts from Chaps. 4 and 5 in order to gain a better understanding of the Neron–Severi lattice of an elliptic surface. This will lead to the announced notion of Mordell–Weil lattice which will be studied in detail in this chapter, but also throughout the remainder of this book.

Mordell-Weil Lattices

Introduction and statement of results Glossary of notation and conventions Gorenstein del Pezzo surfaces Bug-eyed covers Log deformation theory Criteria for log uniruledness Reduction to $\pi^{\text{alg}}_{1}(S^0)=\{1\}$ Flushness and preparation for the hunt Bogomolov bound Riemann Roch and surfaces with small coefficient A partial classification of $K_{T}$-contractions the linear system $\vert K_{S} + A\vert$ Classification of bananas and fences $T_1$ a net $g(A_1)>1$ $A_1$ has a simple cusp $A_1$ has a simple node $A_1$ smooth The smooth banana Proof of (1.1) and corollaries A surface with $\pi^{\text{alg}}_1(S^0)=\{1\}$ but no tiger Tigers, complements and toric pairs Classification of almost all rank one log del Pezzos Appendix: Log terminal surface singularities and adjunction Appendix: Normalisation of an algebraic space Index to obscure or unconventional notation References.

Rational Curves on Quasi-Projective Surfaces

Gorenstein log del Pezzo Surfaces of Rank One

We present a complete list of extremal elliptic K3 surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial (Proposition 4.1 and Theorems 4.3).

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Classification of extremal elliptic {$K3$} surfaces and fundamental groups of open {$K3$} surfaces

For every smooth complex projective variety $W$
 of dimension $d$
 and nonnegative Kodaira dimension, we show the existence of a universal constant $m$
 depending only on $d$
 and two natural invariants of the very general fibres of an Iitaka fibration of $W$
 such that the pluricanonical system $|mK_{W}|$
 defines an Iitaka fibration This is a consequence of a more general result on polarized adjoint divisors In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc

Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

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Logarithmic Enriques surfaces

It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and fibrations.

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De-Qi Zhang

Papers

Gorenstein log del Pezzo Surfaces of Rank One

Classification of extremal elliptic {$K3$} surfaces and fundamental groups of open {$K3$} surfaces

Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

Logarithmic Enriques surfaces

Polarized endomorphisms of complex normal varieties