Abstract
For a canonical threefold of general type, we know that the pluri–canonical map \({\phi_{n}}\) is stably birational for a sufficiently large n. This paper aims to find the lower bound of n for such kind of threefolds with χ = 1. To prove our main result, we will estimate the lower bound of plurigenus.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bombieri E.: Canonical models of surfaces of general type. Publ. I.H.E.S. 42, 171–219 (1973)
Chen M.: Canonical stability of 3-folds of general type with p g ≥ 3. Int. J. Math. 14, 515–528 (2003)
Chen, J.A., Chen, M.: Explicit birational geometry of threefolds of general type. arXiv: 0706.2987v2
Chen J.A., Chen M.: On projective threefolds of general type. Electron. Res. Announc. Math. Sci. 14, 69–73 (2007)
Chen J.A., Hacon C.D.: Pluricanonical systems on irregular 3-folds. Math. Z. 255, 343–355 (2007)
Chen M., Zhang D.Q.: Characterization of the 4-canonical birationality of algebraic threefolds. Math. Z. 258, 565–585 (2008)
Chen, M., Zhu, L.: Projective 3-folds of general type with χ = 1. In: Proceedings of the 4th International Congress of Chinese Mathematicians, vol. II, pp. 314–338. Higher Education Press (2007)
Chen M., Zuo K.: Complex projective threefolds with non-negative canonical Euler-Poincare characteristic. Commun. Anal. Geom. 16, 159–182 (2008)
Chen J.A., Chen M., Zhang D.-Q.: The 5-canonical system on threefolds of general type. J. Reine Angew. Math. 603, 165–181 (2007)
Corti, A., Reid, M.: Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, vol. 281. Cambridge University Press, Cambridge (2000)
Fletcher A.R.: Contributions to Riemann-Roch on projective 3-folds with only canonical singularities and applications. Proc. Symp. Pure Math. 46, 221–231 (1987)
Hacon C.D., McKernan J.: Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166, 1–25 (2006)
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero I. Ann. Math. 79, 109–203; II, ibid. 205–326 (1964)
Iano-Fletcher A.R.: Inverting Reid’s exact plurigenera formula. Math. Ann. 284(4), 617–629 (1989)
Kawamata Y., Matsuda K., Matsuki K.: Introduction to the minimal model problem. Adv. Stud. Pure Math. 10, 283–360 (1987)
Kollár J., Mori S.: Birational geometry of algebraic varieties. Cambridge University Press, Cambridge (1998)
Reid M.: Minimal models of canonical 3-folds. Adv. Stud. Pure Math. 1, 131–180 (1983)
Reid M.: Young person’s guide to canonical singularities. Proc. Symp. Pure Math. 46, 345–414 (1987)
Shin D.K.: On a computation of plurigenera of a canonical threefold. J. Algebra 309, 559–568 (2007)
Takayama S.: Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165, 551–584 (2006)
Tsuji, H.: Pluricanonical systems of projective 3-folds of general type. Preprint. arXiv: 0204096
Zhu L.: The sharp lower bound for the volume of 3-folds of general type with \({{{\chi(\mathcal {O}_{X})=1}}}\). Math. Z. 261(1), 123–141 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
L. Zhu is supported by Fudan Graduate Students’ Innovation Projects (EYH5928004).