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Online ISSN 1534-7486; Print ISSN 1056-3911
Minimal model program for semi-stable threefolds in mixed characteristic
Authors:
Teppei Takamatsu and Shou Yoshikawa
Journal:
J. Algebraic Geom. 32 (2023), 429-476
DOI:
https://doi.org/10.1090/jag/813
Published electronically:
March 24, 2023
Full-text PDF
Abstract | References | Additional Information
Abstract: In this paper, we study the minimal model theory for threefolds in mixed characteristic. As a generalization of a result of Kawamata, we show that the minimal model program (MMP) holds for strictly semi-stable schemes over an excellent Dedekind scheme $V$ of relative dimension two without any assumption on the residue characteristics of $V$. We also prove that we can run a $(K_{X/V}+\Delta )$-MMP over $Z$, where $\pi \colon X \to Z$ is a projective birational morphism of $\mathbb {Q}$-factorial quasi-projective $V$-schemes and $(X,\Delta )$ is a three-dimensional dlt pair with $Exc(\pi ) \subset \lfloor \Delta \rfloor$.
- Bhargav Bhatt, Derived direct summands, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–Princeton University. MR 2753219
- Bhargav Bhatt, On the direct summand conjecture and its derived variant, Invent. Math. 212 (2018), no. 2, 297–317. MR 3787829, DOI 10.1007/s00222-017-0768-7
- Bhargav Bhyatt, Cohen–Macaulayness of absolute integral closures, arXiv:2008.08070, 2020.
- Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek, Globally $+$-regular varieties and the minimal model program for threefolds in mixed characteristic, arXiv:2012.15801, 2020.
- Caucher Birkar, Existence of flips and minimal models for 3-folds in char $p$, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 1, 169–212 (English, with English and French summaries). MR 3465979, DOI 10.24033/asens.2279
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- Caucher Birkar and Joe Waldron, Existence of Mori fibre spaces for 3-folds in $\textrm {char}\,p$, Adv. Math. 313 (2017), 62–101. MR 3649221, DOI 10.1016/j.aim.2017.03.032
- Manuel Blickle, Karl Schwede, and Kevin Tucker, $F$-singularities via alterations, Amer. J. Math. 137 (2015), no. 1, 61–109. MR 3318087, DOI 10.1353/ajm.2015.0000
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
- Sebastien Boucksom, Tommaso de Fernex, and Charles Favre, The volume of an isolated singularity, Duke Math. J. 161 (2012), no. 8, 1455–1520. MR 2931273, DOI 10.1215/00127094-1593317
- Paolo Cascini, Hiromu Tanaka, and Chenyang Xu, On base point freeness in positive characteristic, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 5, 1239–1272 (English, with English and French summaries). MR 3429479, DOI 10.24033/asens.2269
- Bruno Chiarellotto and Christopher Lazda, Combinatorial degenerations of surfaces and Calabi-Yau threefolds, Algebra Number Theory 10 (2016), no. 10, 2235–2266. MR 3582018, DOI 10.2140/ant.2016.10.2235
- Bruno Chiarellotto, Christopher Lazda, and Christian Liedtke, Good reduction of K3 surfaces in equicharacteristic $p$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 1, 483–500. MR 4407199
- Alessio Corti (ed.), Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and its Applications, vol. 35, Oxford University Press, Oxford, 2007. MR 2352762, DOI 10.1093/acprof:oso/9780198570615.001.0001
- Vincent Cossart, Uwe Jannsen, and Shuji Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, arXiv:0905.2191, 2009.
- Vincent Cossart and Olivier Piltant, Resolution of singularities of arithmetical threefolds, J. Algebra 529 (2019), 268–535. MR 3942183, DOI 10.1016/j.jalgebra.2019.02.017
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020, DOI 10.1007/BF02698644
- H. Flenner, L. O’Carroll, and W. Vogel, Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1724388, DOI 10.1007/978-3-662-03817-8
- Osamu Fujino, Special termination and reduction to pl flips, Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl., vol. 35, Oxford Univ. Press, Oxford, 2007, pp. 63–75. MR 2359342, DOI 10.1093/acprof:oso/9780198570615.003.0004
- Yoshinori Gongyo, Yusuke Nakamura, and Hiromu Tanaka, Rational points on log Fano threefolds over a finite field, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 12, 3759–3795. MR 4022715, DOI 10.4171/JEMS/913
- Silvio Greco, Two theorems on excellent rings, Nagoya Math. J. 60 (1976), 139–149. MR 409452, DOI 10.1017/S0027763000017190
- Christopher Hacon and Jakub Witaszek, The minimal model program for threefolds in characteristic five, arXiv:1911.12895, 2019.
- Christopher Hacon and Jakub Witaszek, On the relative minimal model program for threefolds in low characteristics, arXiv:1909.12872, 2019.
- Christopher Hacon and Jakub Witaszek, On the relative minimal model program for fourfolds in positive characteristic, arXiv:2009.02631, 2020.
- Christopher D. Hacon and Chenyang Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), no. 3, 711–744. MR 3327534, DOI 10.1090/S0894-0347-2014-00809-2
- Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
- Robin Hartshorne, Generalized divisors and biliaison, Illinois J. Math. 51 (2007), no. 1, 83–98. MR 2346188
- Uwe Jannsen and Shuji Saito, Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory, J. Algebraic Geom. 21 (2012), no. 4, 683–705. MR 2957692, DOI 10.1090/S1056-3911-2012-00570-0
- Bruce W. Jordan and David R. Morrison, On the Néron models of abelian surfaces with quaternionic multiplication, J. Reine Angew. Math. 447 (1994), 1–22. MR 1263167, DOI 10.1515/crll.1994.447.1
- Yujiro Kawamata, Semistable minimal models of threefolds in positive or mixed characteristic, J. Algebraic Geom. 3 (1994), no. 3, 463–491. MR 1269717
- Yujiro Kawamata, Index 1 covers of log terminal surface singularities, J. Algebraic Geom. 8 (1999), no. 3, 519–527. MR 1689354
- Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286. MR 1680559, DOI 10.2307/121025
- Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 206009, DOI 10.2307/1970447
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- János Kollár, Relative MMP without $\Bbb {Q}$-factoriality, Electron. Res. Arch. 29 (2021), no. 5, 3193–3203. MR 4342251, DOI 10.3934/era.2021033
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- Sándor Kovács, Rational, log canonical, Du Bois singularities. II. Kodaira vanishing and small deformations, Compositio Math. 121 (2000), no. 3, 297–304. MR 1761628, DOI 10.1023/A:1001830707422
- Klaus Künnemann, Projective regular models for abelian varieties, semistable reduction, and the height pairing, Duke Math. J. 95 (1998), no. 1, 161–212. MR 1646554, DOI 10.1215/S0012-7094-98-09505-9
- Christian Liedtke and Yuya Matsumoto, Good reduction of K3 surfaces, Compos. Math. 154 (2018), no. 1, 1–35. MR 3699071, DOI 10.1112/S0010437X17007400
- Linquan Ma and Karl Schwede, Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras, Duke Math. J. 170 (2021), no. 13, 2815–2890. MR 4312190, DOI 10.1215/00127094-2020-0082
- Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek, An analog of adjoint ideals and plt singularities in mixed characteristic, arXiv:1910.14665, 2019.
- Yuya Matsumoto, Good reduction criterion for K3 surfaces, Math. Z. 279 (2015), no. 1-2, 241–266. MR 3299851, DOI 10.1007/s00209-014-1365-8
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- Davesh Maulik, Supersingular K3 surfaces for large primes, Duke Math. J. 163 (2014), no. 13, 2357–2425. With an appendix by Andrew Snowden. MR 3265555, DOI 10.1215/00127094-2804783
- Hiromu Tanaka, Abundance theorem for semi log canonical surfaces in positive characteristic, Osaka J. Math. 53 (2016), no. 2, 535–566. MR 3492812
- Hiromu Tanaka, Semiample perturbations for log canonical varieties over an $F$-finite field containing an infinite perfect field, Internat. J. Math. 28 (2017), no. 5, 1750030, 13. MR 3655076, DOI 10.1142/S0129167X17500306
- Hiromu Tanaka, Behavior of canonical divisors under purely inseparable base changes, J. Reine Angew. Math. 744 (2018), 237–264. MR 3871445, DOI 10.1515/crelle-2015-0111
- Hiromu Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 345–376 (English, with English and French summaries). MR 3795482, DOI 10.5802/aif.3163
- Hiromu Tanaka, Abundance theorem for surfaces over imperfect fields, Math. Z. 295 (2020), no. 1-2, 595–622. MR 4100010, DOI 10.1007/s00209-019-02345-2
- Jakub Witaszek, Relative semiampleness in mixed characteristic, arXiv:2106.06088, 2021.
- Jakub Witaszek, Keel’s base point free theorem and quotients in mixed characteristic, Ann. of Math. (2) 195 (2022), no. 2, 655–705. MR 4387235, DOI 10.4007/annals.2022.195.2.4
Additional Information
Teppei Takamatsu
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Email:
teppeitakamatsu.math@gmail.com
Shou Yoshikawa
Affiliation:
RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), Wako, Saitama 351-0198, Japan
MR Author ID:
1456291
ORCID:
0000-0003-4262-0876
Email:
shou.yoshikawa@riken.jp
Received by editor(s):
April 15, 2021
Received by editor(s) in revised form:
March 4, 2022, April 26, 2022, and July 1, 2022
Published electronically:
March 24, 2023
Additional Notes:
The first author was supported by the Program for Leading Graduate Schools, MEXT, Japan and JSPS KAKENHI Grant number JP19J22795. The second author was supported by the Program for Leading Graduate Schools, MEXT, Japan and JSPS KAKENHI Grant number JP19J22795.
Article copyright:
© Copyright 2023
University Press, Inc.