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EFFECTIVE BOUNDS FOR THE NUMBER OF MMP-SERIES OF A SMOOTH THREEFOLD

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  18 December 2020

DILETTA MARTINELLI
DILETTA MARTINELLI*
Affiliation:
Kortweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, P.O. Box 94248, 1090 GE, Amsterdam, Netherlands e-mail: [email protected]
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Abstract

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We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two.

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14J30: $3$-folds
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 106 - 113
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

References

REFERENCES

Birkar, C., Cascini, P., Hacon, C., and McKernan, J., Existence of minimal models for varieties of general type, J. Amer. Math. Soc. 23(2) (2010), 405–468.CrossRefGoogle Scholar
Cascini, P. and Lazić, V., On the number of a log smooth threefold, Journal de Mathématiques Pures et Appliquées 102(3) (2014), 597–616.CrossRefGoogle Scholar
Cascini, P. and Tasin, L., On the Chern numbers of a smooth threefold, Trans. Am. Math. Soc. 370(11) (2018), 7923–7958.CrossRefGoogle Scholar
Cascini, P. and Zhang, D., Effective ifnite generation for the adjoint rings, Annales de l’Institut Fourier 64(1) (2014), 127–144.CrossRefGoogle Scholar
Debarre, O., Higher-Dimensional algebraic geometry, (Springer Science & Business Media, 2013).Google Scholar
Hacon, C. and McKernan, J., Flips and flops, in Proceedings of the International Congress of Mathematicians, Hyderabad, India (2010), 513–539.Google Scholar
Kawamata, Y., Divisorial contractions to 3-dimensional terminal quotient singularities, in Higher-Dimensional complex varieties (Trento, 1994), (1996), 241–246.Google Scholar
Kawamata, Y., On the cone of divisors of Calabi–Yau fiber spaces, Int. J. Math. 8(5) (1997), 127–144.CrossRefGoogle Scholar
Kawamata, Y. and Matsuki, K., The number of the minimal models for a 3-fold of general type is finite, Math. Ann. 276(4) (1987), 595–598.CrossRefGoogle Scholar
Kollár, J. and Mori, S.. Birational Geometry of algebraic varieties, vol. 134, (Cambridge University Press, 1998).CrossRefGoogle Scholar
Kollár, J., Flops, Nagoya Math. J. 113 (1989), 15–36.CrossRefGoogle Scholar
Kollár, J., Flips and abundance for algebraic threefolds, Ásterisque 211 (1992).Google Scholar
Lesieutre, J., Some constraints on positive entropy automorphisms of smooth threefolds, to appear in Ann. Sci. École Norm. Sup. (arXiv:1503.07834).Google Scholar
Martinelli, D., Scheieder, S., and Tasin, L.. On the number and boundedness of minimal models of general type, Arxiv preprint 1610.08932 (2016).Google Scholar
Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), 133–176.CrossRefGoogle Scholar
Mori, S. and Prokhorov, Y.. General Elephants for threefold extremal contractions with one-dimensional fibers: exceptional case, Arxiv preprint 2002.10693 (2020).Google Scholar
Tziolas, N., Three dimensional divisorial extremal neighborhoods, Math. Annal. 333 (2005), 315–354 CrossRefGoogle Scholar
Reid, M.. Minimal models of canonical threefolds , Adv. Stud. in Pure Math. 1 (1983), 131–180.CrossRefGoogle Scholar
Shokurov, V., The nonvanishing theorem, Math. USSR-Izvestiya 26(3) (1986), 591–604.CrossRefGoogle Scholar