Abstract
We prove an analogue of Fujino and Mori’s “bounding the denominators” (Theorem 3.1 of Fujino and Mori, J Diff Geom 50:167–188, 2000) in the log canonical bundle formula (see also Theorem 8.1 of Prokhorov and Shokurov, J Algebraic Geom 18(1):151–199, 2009) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair (X,Δ) of Kodaira codimension one and dimension at most three such that the coefficients of Δ are in a DCC set \({\mathcal{A}}\), there is a natural number N that depends only on \({\mathcal{A}}\) for which \({\lfloor{N(K_X+\Delta)\rfloor}}\) induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexeev, V.: Boundedness and K 2 for singular surfaces. arXiv:alg-geom/9402007
Alexeev, V., Mori, S.: Bounding singular surfaces of general type. In: Algebra, Arithmetic and Geometry with Applications, pp. 143–174. (2000)
Ambro F.: Shokurov’s boundary property. J. Diff. Geom. 67(2), 309–322 (2004)
Ambro, F.: The adjunction conjecture and its applications. arXiv:math/9903060
Ambro F.: The moduli b-divisor of an lc-trivial fibration. Compos. Math. 141(2), 385–403 (2005)
Fujino O.: Application of Kawamata’s positivity theorem. Proc. Jpn Acad. Ser. A Math. 75(6), 129–171 (1999)
Fujino O., Mori S.: A canonical bundle formula. J. Diff. Geom. 50, 167–188 (2000)
Hacon C., McKernan J.: Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166(1), 1–25 (2006)
Kawamata Y.: Kodaira dimension of certain algebraic fiber spaces. J. Fac. Sci. Univ. Tokyo 30, 1–24 (1983)
Kawamata Y.: Subadjunction of log canonical divisors for a variety of codimension 2. Contemp. Math. 207, 78–88 (1997)
Kawamata Y.: Subadjunction of log canonical divisors II. Am. J. Math. 120, 79–88 (1998)
Kollár, J.: Kodaira’s canonical bundle formula and subadjanction. In: Corti, A., et al. (eds.) Flips for 3-Folds and 4-Folds, pp. 134–162. Oxfrod University Press (2007)
Lazarsfeld R.: Positivity in Algebraic Geometry II. Springer-Verlag, Berlin (2004)
McKernan, J.: Boundedness of log terminal Fano pairs of bounded index. arXiv:math.AG/0205214 v1
Mori S.: Classification of higher dimensional varieties. Proc. Symp. Pure Math. 46, 269–331 (1987)
Pacienza G.: On the uniformity of the Iitaka fibration. Math. Res. Lett. 16(4), 663–681 (2009)
Prokhorov Y.G., Shokurov V.V.: Toward the second main theorem on the complements. J. Algebraic Geom. 18(1), 151–199 (2009)
Ringler, A.: On a conjecture of Hacon and McKernan in dimension three. arXiv:0708.3662
Takayama S.: Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165(3), 551–587 (2006)
Todorov G.: Pluricanonical maps on threefolds of general type. Ann. Inst. Fourier 57(4), 1315–1330 (2007)
Todorov G., Chenyang Xu: Effectiveness of the log Iitaka fibration for 3-folds and 4-folds. Algebra Number Theory 3(6), 697–710 (2009)
Viehweg E., Zhang D.-Q.: Effective Iitaka fibrations. J. Algebraic Geom. 18(4), 711–730 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Todorov, G.T. Effective log Iitaka fibrations for surfaces and threefolds. manuscripta math. 133, 183–195 (2010). https://doi.org/10.1007/s00229-010-0370-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-010-0370-4