*Alessio Corti (ed.)*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.001.0001
- Subject:
- Mathematics, Geometry / Topology

The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with ...
More

The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with negative, zero, and positive curvature, in a similar vein as the geometrization program in topology decomposes a three-manifold into pieces with a standard geometry. The last few years have seen dramatic advances in the minimal model program for higher dimensional algebraic varieties, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalization of the minimal model program and the classification of varieties in all dimensions, comparable to the known results for surfaces and 3-folds. This edited collection of chapters, authored by leading experts, provides a complete and self-contained construction of 3-fold and 4-fold flips, and n-dimensional flips assuming minimal models in dimension n-1. A large part of the text is an elaboration of the work of Shokurov, and a complete and pedagogical proof of the existence of 3-fold flips is presented. The book contains a self-contained treatment of many topics that could only be found, with difficulty, in the specialized literature. The text includes a ten-page glossary.Less

The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with negative, zero, and positive curvature, in a similar vein as the geometrization program in topology decomposes a three-manifold into pieces with a standard geometry. The last few years have seen dramatic advances in the minimal model program for higher dimensional algebraic varieties, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalization of the minimal model program and the classification of varieties in all dimensions, comparable to the known results for surfaces and 3-folds. This edited collection of chapters, authored by leading experts, provides a complete and self-contained construction of 3-fold and 4-fold flips, and n-dimensional flips assuming minimal models in dimension *n-1*. A large part of the text is an elaboration of the work of Shokurov, and a complete and pedagogical proof of the existence of 3-fold flips is presented. The book contains a self-contained treatment of many topics that could only be found, with difficulty, in the specialized literature. The text includes a ten-page glossary.

*Alessio Corti*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational ...
More

This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational functions, and Shokurov's asymptotic saturation property are developed systematically from first principles.Less

This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational functions, and Shokurov's asymptotic saturation property are developed systematically from first principles.

*James McKernan*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.003.0007
- Subject:
- Mathematics, Geometry / Topology

Shokurov proposed the CCS conjecture as an intermediate step leading to his finite generation conjecture implying the existence of pl flips in all dimensions. After the work of Hacon and McKernan, ...
More

Shokurov proposed the CCS conjecture as an intermediate step leading to his finite generation conjecture implying the existence of pl flips in all dimensions. After the work of Hacon and McKernan, the CCS conjecture no longer occupies a central role in the existence theory of flips and the minimal model program, but it is interesting in its own right and it may, in the future, find applications to the theory of Fano varieties. The chapter contains the simplest statement of the CCS conjecture, motivated by the classification of saturated mobile b-divisors on del Pezzo surfaces.Less

Shokurov proposed the CCS conjecture as an intermediate step leading to his finite generation conjecture implying the existence of *pl* flips in all dimensions. After the work of Hacon and McKernan, the CCS conjecture no longer occupies a central role in the existence theory of flips and the minimal model program, but it is interesting in its own right and it may, in the future, find applications to the theory of Fano varieties. The chapter contains the simplest statement of the CCS conjecture, motivated by the classification of saturated mobile b-divisors on del Pezzo surfaces.