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Finite generation of a canonical ring. (English) Zbl 1183.14010

Jerison, David (ed.) et al., Current developments in mathematics, 2007. Somerville, MA: International Press (ISBN 978-1-57146-134-6/pbk). 43-76 (2009).
Motivated by a paper of C. Birkar, P. Cascini, Ch. Hacon and J. McKernan [Existence of minimal models for varieties of log general type, J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019), see also arXiv:math/0610203], the author reviews an algebraic proof of the finite generation theorem, which is one of the most important problems in algebraic geometry. He starts by reviewing the finite generation theorem in the surface case, and explains the numerical geometry and the log theory. The algorithm of the Minimal Model Program (MMP) is explained. He states the main theorems in the proof of the finite generation theorem, and describes essential ideas of the proofs. This is a well-written survey paper.
For the entire collection see [Zbl 1166.00310].

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves

Citations:

Zbl 1210.14019
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