Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access July 7, 2022

Pluricanonical Maps and the Fujita Conjecture

  • Fabrizio Catanese EMAIL logo
From the journal Complex Manifolds

Abstract

We describe examples showing the sharpness of Fujita’s conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.

MSC 2010: 14E05; 14E25; 32Q40

References

[1] Urban Angehrn, Yum Tong Siu, Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308.10.1007/BF01231446Search in Google Scholar

[2] Edoardo Ballico, Roberto Pignatelli, Luca Tasin, Weighted hypersurfaces with either assigned volume or many vanishing plurigenera. Comm. Alg. 41 (2013), 3745–3752.10.1080/00927872.2012.677079Search in Google Scholar

[3] Caucher Birkar, Paolo Cascini, Christopher Derek Hacon, James McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405–468.10.1090/S0894-0347-09-00649-3Search in Google Scholar

[4] Enrico Bombieri, Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. No. 42 (1973), 171–219.10.1007/BF02685880Search in Google Scholar

[5] Jungkai Alfred Chen, Meng Chen, De-Qi Zhang, The 5-canonical system on 3-folds of general type. J. Reine Angew. Math. 603 (2007), 165–181.10.1515/CRELLE.2007.015Search in Google Scholar

[6] Jungkai Alfred Chen, Meng Chen, Zhi Jiang, On 6-canonical map of irregular threefolds of general type. Math. Res. Lett. 20 (2013), no. 1, 19–25.10.4310/MRL.2013.v20.n1.a2Search in Google Scholar

[7] Jungkai Alfred Chen, Meng Chen, Explicit birational geometry of 3-folds and 4-folds of general type, III. Compos. Math. 151 (2015), no. 6, 1041–1082.10.1112/S0010437X14007817Search in Google Scholar

[8] Meng Chen, On minimal 3 -folds of general type with maximal pluricanonical section index. Asian J. Math. 22, No. 2, 257–268 (2018).10.4310/AJM.2018.v22.n2.a3Search in Google Scholar

[9] Jheng-Jie Chen, Jungkai Alfred Chen, Meng Chen, Zhi Jiang, On quint-canonical birationality of irregular threefolds. Proc. Lond. Math. Soc. (3) 122 (2021), no. 2, 234–258.10.1112/plms.12348Search in Google Scholar

[10] Igor Dolgachev, Weighted projective varieties. Group actions and vector fields (Vancouver, B.C., 1981), 34–71, Lecture Notes in Math., 956, Springer, Berlin, 1982.10.1007/BFb0101508Search in Google Scholar

[11] Federigo Enriques, Le Superficie Algebriche. Nicola Zanichelli, Bologna, 1949. xv+464 pp.Search in Google Scholar

[12] Louis Esser, Burt Totaro, Chengxi Wang, Varieties of general type with doubly exponential asymptotics, arXiv:2109.13383.Search in Google Scholar

[13] Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive. Algebraic geometry, Sendai, 1985, 167–178, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.Search in Google Scholar

[14] Christopher Derek Hacon, James McKernan, Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166 (2006), no. 1, 1–25.10.1007/s00222-006-0504-1Search in Google Scholar

[15] Miles Reid, Young person’s guide to canonical singularities. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345–414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.10.1090/pspum/046.1/927963Search in Google Scholar

Received: 2022-06-11
Accepted: 2022-06-26
Published Online: 2022-07-07

© 2022 Fabrizio Catanese, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 29.2.2024 from https://www.degruyter.com/document/doi/10.1515/coma-2021-0136/html
Scroll to top button