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AN EFFECTIVE UPPER BOUND FOR ANTI-CANONICAL VOLUMES OF SINGULAR FANO THREEFOLDS

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  07 March 2024

Chen Jiang  and
Yu Zou Open the ORCID record for Yu Zou [Opens in a new window]
Chen Jiang
Affiliation:
Shanghai Center for Mathematical Sciences & School of Mathematical Sciences, Fudan University, Shanghai, 200438, China (chenjiang@fudan.edu.cn)
Yu Zou*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
*
fishlinazy@tsinghua.edu.cn
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Abstract

For a real number $0<\epsilon <1/3$, we show that the anti-canonical volume of an $\epsilon $-klt Fano $3$-fold is at most $3,200/\epsilon ^4$, and the order $O(1/\epsilon ^4)$ is sharp.

Keywords

Fano threefoldsanti-canonical volumeslog canonical thresholdsboundedness

MSC classification

Primary: 14J45: Fano varieties
Secondary: 14J30: $3$-folds 14J17: Singularities

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 6 , November 2024 , pp. 2631 - 2646
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Alexeev, V., Classification of log canonical surface singularities: Arithmetical proof, in Flips and abundance for algebraic threefolds, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, Astérisque, 211, pp. 47–58 (Societe Mathematique De France, Paris, 1992).Google Scholar
Ambro, F., Variation of log canonical thresholds in linear systems, Int. Math. Res. Not. IMRN 2016(14) (2016), 4418–4448.CrossRefGoogle Scholar
Birkar, C., Anti-pluricanonical systems on Fano varieties, Ann. of Math. (2) 190(2) (2019), 345–463.CrossRefGoogle Scholar
Birkar, C., Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. (2) 193(2) (2021), 347–405.CrossRefGoogle Scholar
Birkar, C., Anticanonical volume of Fano $4$ -folds, in Birational geometry, Kähler–Einstein metrics and Degenerations, Moscow, Shanghai and Pohang, June–November 2019, Springer Proceedings in Mathematics & Statistics, pp. 85–93 (Springer, Switzerland, 2023).CrossRefGoogle Scholar
Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23(2) (2010), 405–468.CrossRefGoogle Scholar
Dolgachev, I. V., Classical Algebraic Geometry: A Modern View (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Jiang, C., Bounding the volumes of singular weak log del Pezzo surfaces, Int. J. Math. 24(13) (2013), 1350110, 27 pp.CrossRefGoogle Scholar
Jiang, C., On birational boundedness of Fano fibrations, Am. J. Math. 140(5) (2018), 1253–1276.CrossRefGoogle Scholar
Jiang, C., Boundedness of anti-canonical volumes of singular log Fano threefolds, Comm. Anal. Geom. 29(7) (2021), 1571–1596.CrossRefGoogle Scholar
Jiang, C. and Zou, Y., An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$ -Fano three-folds, Int. Math. Res. Not. IMRN 2023(11) (2023), 9298–9318.CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, in Cambridge Tracts in Mathematics, 134, pp. i-254 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Lai, C.-J., Bounding volumes of singular Fano threefolds, Nagoya Math. J. 224(1) (2016), 37–73.CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic geometry, I, Classical setting: Line bundles and linear series, in Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics, 48, pp. i–387 (Springer-Verlag, Berlin, 2004).Google Scholar
Mori, S. and Prokhorov, Y. G., On $\mathbb{Q}$ -conic bundles, Publ. Res. Inst. Math. Sci. 44(2) (2008), 315–369.CrossRefGoogle Scholar
Prokhorov, Y. G., Lectures on complements on log surfaces, in MSJ Memoirs, 10, pp. 1–129 (Mathematical Society of Japan, Tokyo, 2001).Google Scholar
Prokhorov, Y. G., The degree of $\mathbb{Q}$ -Fano threefolds, Mat. Sb. 198(11) (2007), 153–174. Translation in Sb. Math. 198(11–12) (2007), 1683–1702.Google Scholar
Zhang, Q., Rational connectedness of log $\mathbb{Q}$ -Fano varieties, J. Reine Angew. Math. 590 (2006), 131–142.Google Scholar