Families of Varieties of General Type

The first complete treatment of the moduli theory of varieties of general type, laying foundations for future research.

János Kollár (Author)

9781009346108, Cambridge University Press

Hardback, published 20 April 2023

466 pages
23.5 x 15.9 x 3.4 cm, 0.88 kg

'This excellent book provides a wealth of examples and technical details for those studying birational geometry and moduli spaces. It completely addresses several state-of-the-art topics in the field, including different stability notions, K-flatness, and subtleties in defining families of stable pairs over an arbitrary base. It will be an essential resource for both those first learning the subject and experts as it moves through history and examples before settling many of the (previously unknown) technicalities needed to define the correct moduli functor.' Kristin DeVleming, University of Massachusetts Amherst

This book establishes the moduli theory of stable varieties, giving the optimal approach to understanding families of varieties of general type. Starting from the Deligne–Mumford theory of the moduli of curves and using Mori's program as a main tool, the book develops the techniques necessary for a theory in all dimensions. The main results give all the expected general properties, including a projective coarse moduli space. A wealth of previously unpublished material is also featured, including Chapter 5 on numerical flatness criteria, Chapter 7 on K-flatness, and Chapter 9 on hulls and husks.

Introduction
Notation
1. History of moduli problems
2. One-parameter families
3. Families of stable varieties
4. Stable pairs over reduced base schemes
5. Numerical flatness and stability criteria
6. Moduli problems with flat divisorial part
7. Cayley flatness
8. Moduli of stable pairs
9. Hulls and husks
10. Ancillary results
11. Minimal models and their singularities
References
Index.

Subject Areas: Algebraic geometry [PBMW], Number theory [PBH]