Duke Mathematical Journal

New outlook on the Minimal Model Program, I

Paolo Cascini and Vladimir Lazić

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Abstract

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.

Article information

Source
Duke Math. J., Volume 161, Number 12 (2012), 2415-2467.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1346936110

Digital Object Identifier
doi:10.1215/00127094-1723755

Mathematical Reviews number (MathSciNet)
MR2976942

Zentralblatt MATH identifier
1261.14007

Subjects
Primary: 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14E99: None of the above, but in this section

Citation

Cascini, Paolo; Lazić, Vladimir. New outlook on the Minimal Model Program, I. Duke Math. J. 161 (2012), no. 12, 2415--2467. doi:10.1215/00127094-1723755. https://projecteuclid.org/euclid.dmj/1346936110


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References

  • [ADHL] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings, preprint, arXiv:1003.4229v2 [math.AG]
  • [BCHM] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.
  • [Bo] N. Bourbaki, Commutative Algebra, Chapters 1–7, reprint of the 1972 ed., Elem. Math. (Berlin), Springer, Berlin, 1989.
  • [CLa] P. Cascini and V. Lazić, The Minimal Model Program revisited, to appear in Contributions to Algebraic Geometry, preprint, arXiv:1202.0738v1 [math.AG]
  • [Co1] A. Corti, “$3$-fold flips after Shokurov” in Flips for $3$-folds and $4$-folds, Oxford Lecture Ser. Math. Appl. 35, Oxford Univ. Press, Oxford, 2007, 18–48.
  • [Co2] A. Corti, “Finite generation of adjoint rings after Lazić: An introduction” in Classification of Algebraic Varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 197–220.
  • [CoLa] A. Corti and V. Lazić, New outlook on the Minimal Model Program, II, to appear in Math. Ann., preprint, arXiv:1005.0614v3 [math.AG]
  • [ELMNP] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), 1701–1734.
  • [FMo] O. Fujino and S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), 167–188.
  • [Fu] W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
  • [HK] C. D. Hacon and S. J. Kovács, Classification of Higher Dimensional Algebraic Varieties, Oberwolfach Semin. 41, Birkhäuser, Basel, 2010.
  • [HM] C. D. Hacon and J. MCKernan, Existence of minimal models for varieties of log general type, II, J. Amer. Math. Soc. 23 (2010), 469–490.
  • [KMo] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
  • [La] V. Lazić, Adjoint rings are finitely generated, preprint, arXiv:0905.2707v3 [math.AG]
  • [Mo] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), 133–176.
  • [N] N. Nakayama, Zariski-decomposition and Abundance, MSJ Mem. 14, Math. Soc. Japan, Tokyo, 2004.
  • [P] M. Păun, Relative critical exponents, non-vanishing and metrics with minimal singularities, Invent. Math. 187 (2012), 195–258.
  • [Sh] V. V. Shokurov, Prelimiting flips (in Russian), Tr. Mat. Inst. Steklova 240 (2003), 82–219; English translation in Proc. Steklov Inst. of Math. 2003 (240), 75–213.
  • [Si1] Y.-T. Siu, Invariance of plurigenera, Invent. Math. 134 (1998), 661–673.
  • [Si2] Y.-T. Siu, Finite generation of canonical ring by analytic method, Sci. China Ser. A 51 (2008), 481–502.