Osaka Journal of Mathematics

The 3-canonical system on 3-folds of general type

Yiqun Zhou

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Let $X$ be a projective minimal Gorenstein 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities. We classify minimal Gorenstein 3-folds of general type according to the birationality of 3-canonical system on $X$.

Article information

Osaka J. Math., Volume 48, Number 1 (2011), 91-98.

First available in Project Euclid: 22 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E05: Rational and birational maps 14J10: Families, moduli, classification: algebraic theory 14J30: $3$-folds [See also 32Q25]


Zhou, Yiqun. The 3-canonical system on 3-folds of general type. Osaka J. Math. 48 (2011), no. 1, 91--98.

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  • W. Barth, K. Hulek, C. Peters and A. Van de Ven: Compact Complex Surfaces, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Springer, Berlin, 2004.
  • E. Bombieri: Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219.
  • M. Chen: Canonical stability in terms of singularity index for algebraic threefolds, Math. Proc. Cambridge Philos. Soc. 131 (2001), 241–264.
  • M. Chen: Canonical stability of 3-folds of general type with $p_{g} \geq 3$, Internat. J. Math. 14 (2003), 515–528.
  • J.A. Chen, M. Chen and D.-Q. Zhang: The 5-canonical system on 3-folds of general type, J. Reine Angew. Math. 603 (2007), 165–181.
  • M. Chen and D.-Q. Zhang: Characterization of the 4-canonical birationality of algebraic threefolds, Math. Z. 258 (2008), 565–585.
  • Y. Kawamata, K. Matsuda and K. Matsuki: Introduction to the minimal model problem; in Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 283–360.
  • Y. Kawamata: A generalization of Kodaira–Ramanujam's vanishing theorem, Math. Ann. 261 (1982), 43–46.
  • M. Kobayashi: On Noether's inequality for threefolds, J. Math. Soc. Japan 44 (1992), 145–156.
  • I. Reider: Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), 309–316.
  • E. Viehweg: Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8.
  • L. Zhu: The generic finiteness of the $m$-canonical map for 3-folds of general type, Osaka J. Math. 42 (2005), 873–884.