Abstract
It was proved by Chen and Chen that a terminal Fano 3-fold X satisfies \((-K_X)^3\ge \frac{1}{330}\). We show that a non-rational \(\mathbb {Q}\)-factorial terminal Fano 3-fold X with \(\rho (X)=1\) and \((-K_X)^3=\frac{1}{330}\) is a weighted hypersurface of degree 66 in \(\mathbb {P}(1,5,6,22,33)\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alexeev, V.: General elephants of \(\mathbb{Q} \)-Fano \(3\)-folds. Compositio Math. 91(1), 91–116 (1994)
Buckley, A., Reid, M., Zhou, S.: Ice cream and orbifold Riemann-Roch. Izv. Math. 77(3), 461–486 (2013)
Cheltsov, I., Park, J.: Birationally rigid Fano threefold hypersurfaces. Mem. Amer. Math. Soc. 246(1167), v+117 (2017)
Chen, J.A., Chen, M.: An optimal boundedness on weak \(\mathbb{Q} \)-Fano \(3\)-folds. Adv. Math. 219(6), 2086–2104 (2008)
Chen, M., Jiang, C.: On the anti-canonical geometry of \(\mathbb{Q} \)-Fano threefolds. J. Differ. Geom. 104(1), 59–109 (2016)
Corti, A., Pukhlikov, A., Reid, M.: Fano \(3\)-fold hypersurfaces, Explicit birational geometry of \(3\)-folds, London Mathematical Society. Lecture Note Series, vol. 281, pp. 175–258. Cambridge University Press, Cambridge (2000)
de Fernex, T., Ein, L., Mustaţă, M.: Log canonical thresholds on varieties with bounded singularities. Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., pp. 221–257 Zürich (2011)
Dolgachev, I.: Weighted projective varieties. Group Actions and Vector Fields (Vancouver, B.C., 1981). Lecture Notes in Mathematics, vol. 956, pp. 34–71. Springer, Berlin (1982)
Iano-Fletcher, A.R.: Working with weighted complete intersections, Explicit birational geometry of \(3\)-folds. London Mathematical Society. Lecture Note Series, vol. 281, pp. 101–173. Cambridge University Press, Cambridge (2000)
Jiang, C.: On boundedness of volumes and birationality in birational geometry, Ph.D. thesis, University of Tokyo, vi+115 pp (2015). https://chenjiangfudan.github.io/home/thesis.pdf
Jiang, C., Zou, Y.: An effective upper bound for anti-canonical volumes of canonical \(\mathbb{Q}\)-Fano threefolds, to appear in Int. Math. Res. Not. arXiv:2107.01037
Namikawa, Y.: Smoothing Fano \(3\)-folds. J. Algebraic Geom. 6(2), 307–324 (1997)
Prokhorov, Y.G.: The degree of \(\mathbb{Q}\)-Fano threefolds. Mat. Sb. 198 (11), 153–174 (2007). Translation in Sb. Math. 198(11–12), 1683–1702 (2007)
Reid, M.: Young person’s guide to canonical singularities. Algebraic Geometry, Bowdoin (1985) (Brunswick, Maine, 1985). Proceedings of Symposium Pure Mathematics, vol. 46, Part 1, pp. 345–414. American Mathematical Society, Providence, RI (1987)
Acknowledgements
The author was supported by National Key Research and Development Program of China (Grant No. 2020YFA0713200) and NSFC for Innovative Research Groups (Grant No. 12121001). This paper was written during the author’s visit to Xiamen University in July 2021, and he is grateful for the hospitality and support of Wenfei Liu and XMU. The author would like to thank Yifei Chen and the referees for useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Jiang, C. (2023). Characterizing Terminal Fano Threefolds with the Smallest Anti-canonical Volume. In: Cheltsov, I., Chen, X., Katzarkov, L., Park, J. (eds) Birational Geometry, Kähler–Einstein Metrics and Degenerations. BGKEMD BGKEMD BGKEMD 2019 2019 2019. Springer Proceedings in Mathematics & Statistics, vol 409. Springer, Cham. https://doi.org/10.1007/978-3-031-17859-7_18
Download citation
DOI: https://doi.org/10.1007/978-3-031-17859-7_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-17858-0
Online ISBN: 978-3-031-17859-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)