## Birational geometry old and new

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**46**(2009), 99-123 Request permission

## Abstract:

A classical problem in algebraic geometry is to describe quantities that are invariants under birational equivalence as well as to determine some convenient birational model for each given variety, a minimal model. One such quantity is the ring of objects which transform like a tensor power of a differential of top degree, known as the canonical ring. The histories of the existence of minimal models and the finite generation of the canonical ring are intertwined; minimal models and canonical rings constitute the major building blocks for the birational classification of algebraic varieties. In this paper we will discuss some of the ideas involved, recent advances on the existence of minimal models, some applications, and the (algebraic-geometric proof of the) finite generation of the canonical ring. These results have been long standing conjectures in algebraic geometry.## References

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## Additional Information

**Antonella Grassi**- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- Email: grassi@math.upenn.edu
- Received by editor(s): June 8, 2008
- Published electronically: October 27, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc.
**46**(2009), 99-123 - MSC (2000): Primary 14E30; Secondary 14J99
- DOI: https://doi.org/10.1090/S0273-0979-08-01233-0
- MathSciNet review: 2457073