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Birational geometry old and new


Author: Antonella Grassi
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
Published electronically: October 27, 2008
MathSciNet review: 2457073
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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.


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Additional Information

Antonella Grassi
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: grassi@math.upenn.edu

DOI: https://doi.org/10.1090/S0273-0979-08-01233-0
Keywords: Algebraic geometry
Received by editor(s): June 8, 2008
Published electronically: October 27, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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