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ISSN 1088-9485(e) ISSN 0273-0979(p)

Birational geometry old and new

Author(s): Antonella Grassi
Journal: Bull. Amer. Math. Soc. 46 (2009), 99-123.
MSC (2000): Primary 14E30; Secondary 14J99
Posted: October 27, 2008
MathSciNet review: 2457073
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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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