Algebra & Number Theory

Effectiveness of the log Iitaka fibration for 3-folds and 4-folds

Gueorgui Todorov and Chenyang Xu

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Abstract

We prove the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension 4. In particular, we finish the proof of effective log Iitaka fibration in dimension two. Also, we show that for the log Iitaka fibration, if the fiber is of dimension two, the denominator of the moduli part is bounded.

Article information

Source
Algebra Number Theory, Volume 3, Number 6 (2009), 697-710.

Dates
Received: 20 January 2009
Revised: 9 July 2009
Accepted: 17 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797470

Digital Object Identifier
doi:10.2140/ant.2009.3.697

Mathematical Reviews number (MathSciNet)
MR2579391

Zentralblatt MATH identifier
1184.14023

Subjects
Primary: 14E05: Rational and birational maps
Secondary: 14J35: $4$-folds 14J30: $3$-folds [See also 32Q25]

Keywords
Iitaka fibration boundedness

Citation

Todorov, Gueorgui; Xu, Chenyang. Effectiveness of the log Iitaka fibration for 3-folds and 4-folds. Algebra Number Theory 3 (2009), no. 6, 697--710. doi:10.2140/ant.2009.3.697. https://projecteuclid.org/euclid.ant/1513797470


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References

  • V. Alexeev, “Boundedness and $K\sp 2$ for log surfaces”, Internat. J. Math. 5:6 (1994), 779–810.
  • V. Alexeev and S. Mori, “Bounding singular surfaces of general type”, pp. 143–174 in Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), edited by C. Christensen et al., Springer, Berlin, 2004.
  • C. Birkar, P. Cascini, C. Hacon, and J. M$^{\mathrm c}$Kernan, “Existence of minimal models for varieties of log general type”, preprint, 2006.
  • O. Fujino and S. Mori, “A canonical bundle formula”, J. Differential Geom. 56:1 (2000), 167–188.
  • C. D. Hacon and J. M$^{\mathrm c}$Kernan, “Boundedness of pluricanonical maps of varieties of general type”, Invent. Math. 166:1 (2006), 1–25.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • Y. Kawamata, “On the plurigenera of minimal algebraic $3$-folds with $K\mathop{\rlap{\raise3.25pt\hbox{$\sim$}}\rlap{$\approx$}\phantom{\approx} }0$”, Math. Ann. 275:4 (1986), 539–546.
  • Y. Kawamata, “Subadjunction of log canonical divisors, II”, Amer. J. Math. 120:5 (1998), 893–899.
  • J. Kollár, “Log surfaces of general type; some conjectures”, pp. 261–275 in Classification of algebraic varieties (L'Aquila, 1992), edited by C. Ciliberto et al., Contemp. Math. 162, Amer. Math. Soc., Providence, RI, 1994.
  • J. Kollár, “Kodaira's canonical bundle formula and adjunction”, pp. 134–162 in Flips for 3-folds and 4-folds, edited by A. Corti, Oxford Lecture Ser. Math. Appl. 35, Oxford Univ. Press, Oxford, 2007.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, New York, 1998.
  • R. Lazarsfeld, Positivity in algebraic geometry, I, Ergebnisse der Math. (3) 48, Springer, Berlin, 2004.
  • G. Pacienza, “On the uniformity of the Iitaka fibration”, preprint, 2007. To appear in Math. Research Letters.
  • Y. Prokhorov and V. Shokurov, “Towards the second main theorem on complements”, J. Algebraic Geom. 18 (2009), 151–199.
  • A. Ringler, “On a conjecture of Hacon and McKernan in dimension three”, preprint, 2007.
  • V. Shokurov, “3-fold log flips”, Izv. Russ. A. N. Ser. Mat. 56:1 (1992), 105–203. In Russian; translated in Russ. Acad. Sci., Izv., Math. 40:1 (1993), 95–202.
  • S. Takayama, “Pluricanonical systems on algebraic varieties of general type”, Invent. Math 165:3 (2006), 551–587.
  • G. Todorov, “Effective log Iitaka fibrations for surfaces and threefolds”, preprint, 2008.
  • E. Viehweg and D. Zhang, “Effective Iitaka fibrations”, preprint, 2007. To appear in J. Alg. Geom.