Hacon, Christopher D.; McKernan, James Boundedness of pluricanonical maps of varieties of general type. (English) Zbl 1121.14011 Invent. Math. 166, No. 1, 1-25 (2006). In this paper the authors prove the following result: for any positive integer \(n\) there exists an integer \(r_n\) such that if \(X\) is a smooth projective variety of general type and dimension \(n\), then the \(r\)-canonical map of \(X\) is birational onto its image for all \(r\geq r_n\). Reviewer: Ivan Cheltsov (Edinburgh) Cited in 18 ReviewsCited in 86 Documents MSC: 14E30 Minimal model program (Mori theory, extremal rays) 14E05 Rational and birational maps Keywords:pluricanonical map; variety of general type; birational map × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Ambro, F.: The locus of log canonical singularities. arXiv:math.AG/9806067 [2] Benveniste, X.: Sur les applications pluricanoniques des variétés de type très général en dimension 3. Am. J. Math. 108, 433–449 (1986) · Zbl 0601.14035 · doi:10.2307/2374679 [3] Bombieri, E.: The pluricanonical map of a complex surface. In: Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, MD 1970), pp. 35–87. Berlin: Springer 1970 · Zbl 0213.47601 [4] Campana, F.: Orbifolds, special varieties and classification theory. Ann. Inst. Fourier 54, 499–630 (2004) · Zbl 1062.14014 [5] Chen, M.: The relative pluricanonical stability for 3-folds of general type. Proc. Am. Math. Soc. 129, 1927–1937 (2001) (electronic) · Zbl 0966.14007 · doi:10.1090/S0002-9939-00-05870-6 [6] Hacon, C., McKernan, J.: On Shokurov’s rational connectedness conjecture. math.AG/0504330 · Zbl 1128.14028 [7] Hanamura, M.: Pluricanonical maps of minimal 3-folds. Proc. Japan Acad., Ser. A, Math. Sci. 61, 116–118 (1985) · Zbl 0598.14032 · doi:10.3792/pjaa.61.116 [8] Iano-Fletcher, A.R.: Working with weighted complete intersections. In: Explicit Birational Geometry of 3-folds. Lond. Math. Soc. Lect. Note Ser., vol. 281, pp. 101–173. Cambridge: Cambridge Univ. Press (2000) · Zbl 0960.14027 [9] Kawamata, Y.: On Fujita’s freeness conjecture for 3-folds and 4-folds. Math. Ann. 308, 491–505 (1997) · Zbl 0909.14001 · doi:10.1007/s002080050085 [10] Kawamata, Y.: On the extension problem of pluricanonical forms. In: Algebraic Geometry: Hirzebruch 70 (Warsaw 1998), Contemp. Math., vol. 241, pp. 193–207. Providence, RI: Am. Math. Soc. 1999 · Zbl 0972.14005 [11] Kollár, J.: Higher direct images of dualizing sheaves I. Ann. Math. 123, 11–42 (1986) · Zbl 0598.14015 · doi:10.2307/1971351 [12] Kollár, J. et al: Flips and abundance for algebraic threefolds. Paris: Société Mathématique de France 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991. Astérisque 211 (1992) [13] Kollár, J., Mori, S.: Birational geometry of algebraic varieties. Cambr. Tracts Math., vol. 134. Cambridge University Press 1998 · Zbl 0926.14003 [14] Lazarsfeld, R.: Positivity in algebraic geometry, II. Ergeb. Math. Grenzgeb, 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49. Berlin: Springer 2004. Positivity for vector bundles, and multiplier ideals [15] Lu, S.: A refined kodaira dimension and its canonical fibration. arXiv:math.AG/0211029 [16] Luo, T.: Plurigenera of regular threefolds. Math. Z. 217, 37–46 (1994) · Zbl 0808.14029 · doi:10.1007/BF02571932 [17] Luo, T.: Global holomorphic 2-forms and pluricanonical systems on threefolds. Math. Ann. 318, 707–730 (2000) · Zbl 1005.14016 · doi:10.1007/s002080000136 [18] Maehara, K.: A finiteness property of varieties of general type. Math. Ann. 262, 101–123 (1983) · doi:10.1007/BF01474173 [19] Matsuki, K.: On pluricanonical maps for 3-folds of general type. J. Math. Soc. Japan 38, 339–359 (1986) · doi:10.2969/jmsj/03820339 [20] Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134, 661–673 (1998) · Zbl 0955.32017 · doi:10.1007/s002220050276 [21] Szabó, E.: Divisorial log terminal singularities. J. Math. Sci., Tokyo 1, 631–639 (1994) · Zbl 0835.14001 [22] Tsuji, H.: Pluricanonical systems of projective varieties of general type. arXiv:math.AG/09909021 · Zbl 1186.14043 [23] Tsuji, H.: Subadjunction theorem for pluricanonical divisors. arXiv:math.AG/0111311 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.