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doi: 10.3934/era.2021033
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## Relative mmp without $\mathbb{Q}$-factoriality

 Princeton University, Princeton NJ 08544-1000, USA

Received  December 2020 Revised  February 2021 Early access April 2021

Fund Project: Partial financial support was provided by the NSF under grant number DMS-1901855

We consider the minimal model program for varieties that are not $\mathbb{Q}$-factorial. We show that, in many cases, its steps are simpler than expected. The main applications are to log terminal singularities, removing the earlier $\mathbb{Q}$-factoriality assumption from several theorems of Hacon-Witaszek and de Fernex-Kollár-Xu.

Citation: János Kollár. Relative mmp without $\mathbb{Q}$-factoriality. Electronic Research Archive, doi: 10.3934/era.2021033
##### References:
 [1] Flips and Abundance for Algebraic Threefolds, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211, (1992). Google Scholar [2] H. Ahmadinezhad, M. Fedorchuk and I. Krylov, Stability of fibrations over one-dimensional bases, 2019, arXiv: 1912.08779. Google Scholar [3] V. Alexeev, Moduli of Weighted Hyperplane Arrangements, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015, Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrì and Paolo Stellari. doi: 10.1007/978-3-0348-0915-3.  Google Scholar [4] E. Arvidsson, F. Bernasconi and J. Lacini, On the Kawamata-Viehweg vanishing for log del Pezzo surfaces in positive characteristic, 2020, arXiv: 2006.03571. Google Scholar [5] F. Bernasconi and J. Kollár, Vanishing theorems for threefolds in characteristic $p>5$, 2020, arXiv: 2012.08343. Google Scholar [6] B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron and J. Witaszek, Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic, 2020, arXiv: 2012.15801. Google Scholar [7] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468.  doi: 10.1090/S0894-0347-09-00649-3.  Google Scholar [8] J.-F. Boutot, Schéma de Picard Local, vol. 632 of Lecture Notes in Mathematics, Springer, Berlin, 1978.  Google Scholar [9] J. Carvajal-Rojas and A. Stäbler, On the local étale fundamental group of KLT threefold singularities, 2020, arXiv: 2004.07628, With an appendix by János Kollár. Google Scholar [10] V. Cossart and O. Piltant, Resolution of singularities of arithmetical threefolds, J. Algebra, 529 (2019), 268-535.  doi: 10.1016/j.jalgebra.2019.02.017.  Google Scholar [11] T. de Fernex, J. Kollár and C. Xu, The dual complex of singularities, in Higher Dimensional Algebraic Geometry--in Honour of Professor Yujiro Kawamata's Sixtieth Birthday, vol. 74 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, (2017), 103-129. doi: 10.2969/aspm/07410103.  Google Scholar [12] O. Fujino, Foundations of the Minimal Model Program, vol. 35 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2017.  Google Scholar [13] H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math., 11 (1970), 263-292.  doi: 10.1007/BF01403182.  Google Scholar [14] C. Hacon and J. Witaszek, On the rationality of Kawamata log terminal singularities in positive characteristic, Algebr. Geom., 6 (2019), 516-529.  doi: 10.14231/ag-2019-023.  Google Scholar [15] C. Hacon and J. Witaszek, On the relative minimal model program for threefolds in low characteristics, 2019, arXiv: 1909.12872. Google Scholar [16] C. Hacon and J. Witaszek, On the relative minimal model program for fourfolds in positive characteristic, 2020, arXiv: 2009.02631. Google Scholar [17] J. Han, J. Liu and V. V. Shokurov, Acc for minimal log discrepancies of exceptional singularities, 2019, arXiv: 1903.04338. Google Scholar [18] J. Kollár, Lectures on Resolution of Singularities, vol. 166 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2007.  Google Scholar [19] J. Kollár, A local version of the Kawamata-Viehweg vanishing theorem, Pure Appl. Math. Q., 7 (2011), 1477-1494.   Google Scholar [20] J. Kollár, Singularities of the Minimal Model Program, vol. 200 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2013, With a collaboration of Sándor Kovács. doi: 10.1017/CBO9781139547895.  Google Scholar [21] J. Kollár, Appendix to "On the local tale fundamental group of KLT threefold singularities", 2020. Google Scholar [22] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. doi: 10.1017/CBO9780511662560.  Google Scholar [23] J. Kollár and J. Witaszek, Resolution and alteration with ample exceptional divisor, 2021, arXiv: 2102.03162. Google Scholar [24] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 195-279, URL http://www.numdam.org/item?id=PMIHES_1969__36__195_0.  Google Scholar [25] Y. Odaka and C. Xu, Log-canonical models of singular pairs and its applications, Math. Res. Lett., 19 (2012), 325-334.  doi: 10.4310/MRL.2012.v19.n2.a5.  Google Scholar [26] A. J. Parameswaran and V. Srinivas, A variant of the Noether-Lefschetz theorem: Some new examples of unique factorisation domains, J. Algebraic Geom., 3 (1994), 81-115.   Google Scholar [27] A. J. Parameswaran and D. van Straten, Algebraizations with minimal class group, Internat. J. Math., 4 (1993), 989-996.  doi: 10.1142/S0129167X93000455.  Google Scholar [28] M. Reid, Young person's guide to canonical singularities, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), vol. 46 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1987,345-414.  Google Scholar [29] V. V. Shokurov, Three-dimensional log perestroikas, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 56 (1992), 105-203.  doi: 10.1070/IM1993v040n01ABEH001862.  Google Scholar [30] T. Stacks project authors, The stacks project, https://stacks.math.columbia.edu, 2021. Google Scholar [31] H. Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble), 68 (2018), 345-376.  doi: 10.5802/aif.3163.  Google Scholar [32] M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math., 219 (2008), 488-522.  doi: 10.1016/j.aim.2008.05.006.  Google Scholar [33] D. Villalobos-Paz, 2021, (in preparation). Google Scholar [34] J. Witaszek, Non Q-factorial flips in dimension three, 2020, Available from: http://www-personal.umich.edu/ jakubw/Non-Q-factorial-flips-in-dimension-three.pdf. Google Scholar [35] C. Xu, Finiteness of algebraic fundamental groups, Compos. Math., 150 (2014), 409-414.  doi: 10.1112/S0010437X13007562.  Google Scholar

show all references

##### References:
 [1] Flips and Abundance for Algebraic Threefolds, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211, (1992). Google Scholar [2] H. Ahmadinezhad, M. Fedorchuk and I. Krylov, Stability of fibrations over one-dimensional bases, 2019, arXiv: 1912.08779. Google Scholar [3] V. Alexeev, Moduli of Weighted Hyperplane Arrangements, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015, Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrì and Paolo Stellari. doi: 10.1007/978-3-0348-0915-3.  Google Scholar [4] E. Arvidsson, F. Bernasconi and J. Lacini, On the Kawamata-Viehweg vanishing for log del Pezzo surfaces in positive characteristic, 2020, arXiv: 2006.03571. Google Scholar [5] F. Bernasconi and J. Kollár, Vanishing theorems for threefolds in characteristic $p>5$, 2020, arXiv: 2012.08343. Google Scholar [6] B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron and J. Witaszek, Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic, 2020, arXiv: 2012.15801. Google Scholar [7] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468.  doi: 10.1090/S0894-0347-09-00649-3.  Google Scholar [8] J.-F. Boutot, Schéma de Picard Local, vol. 632 of Lecture Notes in Mathematics, Springer, Berlin, 1978.  Google Scholar [9] J. Carvajal-Rojas and A. Stäbler, On the local étale fundamental group of KLT threefold singularities, 2020, arXiv: 2004.07628, With an appendix by János Kollár. Google Scholar [10] V. Cossart and O. Piltant, Resolution of singularities of arithmetical threefolds, J. Algebra, 529 (2019), 268-535.  doi: 10.1016/j.jalgebra.2019.02.017.  Google Scholar [11] T. de Fernex, J. Kollár and C. Xu, The dual complex of singularities, in Higher Dimensional Algebraic Geometry--in Honour of Professor Yujiro Kawamata's Sixtieth Birthday, vol. 74 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, (2017), 103-129. doi: 10.2969/aspm/07410103.  Google Scholar [12] O. Fujino, Foundations of the Minimal Model Program, vol. 35 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2017.  Google Scholar [13] H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math., 11 (1970), 263-292.  doi: 10.1007/BF01403182.  Google Scholar [14] C. Hacon and J. Witaszek, On the rationality of Kawamata log terminal singularities in positive characteristic, Algebr. Geom., 6 (2019), 516-529.  doi: 10.14231/ag-2019-023.  Google Scholar [15] C. Hacon and J. Witaszek, On the relative minimal model program for threefolds in low characteristics, 2019, arXiv: 1909.12872. Google Scholar [16] C. Hacon and J. Witaszek, On the relative minimal model program for fourfolds in positive characteristic, 2020, arXiv: 2009.02631. Google Scholar [17] J. Han, J. Liu and V. V. Shokurov, Acc for minimal log discrepancies of exceptional singularities, 2019, arXiv: 1903.04338. Google Scholar [18] J. Kollár, Lectures on Resolution of Singularities, vol. 166 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2007.  Google Scholar [19] J. Kollár, A local version of the Kawamata-Viehweg vanishing theorem, Pure Appl. Math. Q., 7 (2011), 1477-1494.   Google Scholar [20] J. Kollár, Singularities of the Minimal Model Program, vol. 200 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2013, With a collaboration of Sándor Kovács. doi: 10.1017/CBO9781139547895.  Google Scholar [21] J. Kollár, Appendix to "On the local tale fundamental group of KLT threefold singularities", 2020. Google Scholar [22] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. doi: 10.1017/CBO9780511662560.  Google Scholar [23] J. Kollár and J. Witaszek, Resolution and alteration with ample exceptional divisor, 2021, arXiv: 2102.03162. Google Scholar [24] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 195-279, URL http://www.numdam.org/item?id=PMIHES_1969__36__195_0.  Google Scholar [25] Y. Odaka and C. Xu, Log-canonical models of singular pairs and its applications, Math. Res. Lett., 19 (2012), 325-334.  doi: 10.4310/MRL.2012.v19.n2.a5.  Google Scholar [26] A. J. Parameswaran and V. Srinivas, A variant of the Noether-Lefschetz theorem: Some new examples of unique factorisation domains, J. Algebraic Geom., 3 (1994), 81-115.   Google Scholar [27] A. J. Parameswaran and D. van Straten, Algebraizations with minimal class group, Internat. J. Math., 4 (1993), 989-996.  doi: 10.1142/S0129167X93000455.  Google Scholar [28] M. Reid, Young person's guide to canonical singularities, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), vol. 46 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1987,345-414.  Google Scholar [29] V. V. Shokurov, Three-dimensional log perestroikas, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 56 (1992), 105-203.  doi: 10.1070/IM1993v040n01ABEH001862.  Google Scholar [30] T. Stacks project authors, The stacks project, https://stacks.math.columbia.edu, 2021. Google Scholar [31] H. Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble), 68 (2018), 345-376.  doi: 10.5802/aif.3163.  Google Scholar [32] M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math., 219 (2008), 488-522.  doi: 10.1016/j.aim.2008.05.006.  Google Scholar [33] D. Villalobos-Paz, 2021, (in preparation). Google Scholar [34] J. Witaszek, Non Q-factorial flips in dimension three, 2020, Available from: http://www-personal.umich.edu/ jakubw/Non-Q-factorial-flips-in-dimension-three.pdf. Google Scholar [35] C. Xu, Finiteness of algebraic fundamental groups, Compos. Math., 150 (2014), 409-414.  doi: 10.1112/S0010437X13007562.  Google Scholar
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