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2024, Volume 32, Issue 5: 3635-3699. doi: 10.3934/era.2024167
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Research article

Minimal resolutions of threefolds

  • School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
  • Received: 30 October 2023 Revised: 19 March 2024 Accepted: 30 April 2024 Published: 04 June 2024

  • We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.

    Citation: Hsin-Ku Chen. Minimal resolutions of threefolds[J]. Electronic Research Archive, 2024, 32(5): 3635-3699. doi: 10.3934/era.2024167

    Related Papers:

  • We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.


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    [1] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford University Press, Oxford, 2006.
    [2] J. A. Chen, Explicit resolution of three dimensional terminal singularities, in Minimal Models and Extremal Rays (Kyoto, 2011), Mathematical Society of Japan, (2016), 323–360.
    [3] J. A. Chen, C. D. Hacon, Factoring 3-fold flips and divisorial contractions to curves, J. Reine Angew. Math., 657 (2011), 173–197. https://doi.org/10.1515/crelle.2011.056 doi: 10.1515/crelle.2011.056
    [4] H. K. Chen, On the Nash problem for terminal threefolds of type $cA/r$, Int. J. Math., 34 (2023), 2350055.
    [5] M. Reid, Minimal model of canonical 3-folds, Adv. Stud. Pure Math., 1 (1983), 131–180.
    [6] S. Mori, On 3-dimensional terminal singularities, Nagoya Math. J., 98 (1985), 43–66. https://doi.org/10.1017/S0027763000021358 doi: 10.1017/S0027763000021358
    [7] M. Reid, Young person's guide to canonical singularities, Proc. Symp. Pure Math., 46 (1987), 345–414.
    [8] Y. Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities, in Higher Dimensional Complex Varieties, De Gruyter, (1996), 241–246. https://doi.org/10.1515/9783110814736.241
    [9] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities, Publ. Res. Inst. Math. Sci., 35 (1999), 515–570. https://doi.org/10.2977/prims/1195143612 doi: 10.2977/prims/1195143612
    [10] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities, II, Publ. Res. Inst. Math. Sci., 36 (2000), 423–456. https://doi.org/10.2977/prims/1195142953 doi: 10.2977/prims/1195142953
    [11] T. Hayakawa, Divisorial contractions to 3-dimensional terminal singularities with discrepancy one, J. Math. Soc. Japan, 57 (2005), 651–668. https://doi.org/10.2969/jmsj/1158241927 doi: 10.2969/jmsj/1158241927
    [12] M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math., 145 (2001), 105–119. https://doi.org/10.1007/s002220100144 doi: 10.1007/s002220100144
    [13] M. Kawakita, Three-fold divisorial contractions to singularities of higher indices, Duke Math. J., 130 (2005), 57–126. https://doi.org/10.1215/S0012-7094-05-13013-7 doi: 10.1215/S0012-7094-05-13013-7
    [14] M. Kawakita, Supplement to classification of three-fold divisorial contractions, Nagoya Math. J., 208 (2012), 67–73. https://doi.org/10.1215/00277630-1548493 doi: 10.1215/00277630-1548493
    [15] Y. Yamamoto, Divisorial contractions to cDV points with discrepancy $>1$, Kyoto J. Math., 58 (2018), 529–567.
    [16] J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties, Cambridge University Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511662560
    [17] J. Kollár, Flops, Nagoya Math. J., 113 (1989), 15–36.
    [18] J. Kollár, Flips, flops, minimal models, etc, Surv. Diff. Geom., 1 (1991), 113–199.
    [19] H. K. Chen, On the factorization of three-dimensional terminal flops, preprint, arXiv: 2004.12711.
    [20] J. Kollár, S. Mori, Classification of three-dimensional terminal flips, J. Amer. Math. Soc., 5 (1992), 533–703. https://doi.org/10.2307/2152704 doi: 10.2307/2152704
    [21] X. Benveniste, Sur le cone des 1-cycles effectifs en dimension 3, Math. Ann., 272 (1985), 257–265.
    [22] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math., 116 (1982), 113–176.
    [23] Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci., 44 (2008), 419–423.
    [24] H. K. Chen, Betti numbers in the three-dimensional minimal model program, Bull. London Math. Soc., 51 (2019), 563–576. https://doi.org/10.1007/s002220100185 doi: 10.1007/s002220100185
    [25] J. M. Johnson, J. Kollár, Arc spaces of $cA$-type singularities, J. Sing., 7 (2013), 238–252. http://doi.org/10.5427/jsing.2013.7m doi: 10.5427/jsing.2013.7m
    [26] A. Kuznetsov, Semiorthogonal decompositions in algebraic geometry, in Proceedings of the International Congress of Mathematicians-Seoul 2014, (2014), 635–660.
    [27] D. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math., 41 (1993), 133–141.
    [28] T. Bridgeland, Flops and derived categories, Invent. Math., 147 (2002), 613–632. https://doi.org/10.1007/s002220100185 doi: 10.1007/s002220100185
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  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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