Geodesic
On the relative minimal model program for fourfolds in positive and mixed characteristic
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic $p>5$: for contractions to ${\mathbb {Q}}$-factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi–Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions. 
@article{10_1017_fmp_2023_6,
     author = {Christopher Hacon and Jakub Witaszek},
     title = {On the relative minimal model program for fourfolds in positive and mixed characteristic},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {11},
     year = {2023},
     doi = {10.1017/fmp.2023.6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.6/}
}
TY  - JOUR
AU  - Christopher Hacon
AU  - Jakub Witaszek
TI  - On the relative minimal model program for fourfolds in positive and mixed characteristic
JO  - Forum of Mathematics, Pi
PY  - 2023
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.6/
DO  - 10.1017/fmp.2023.6
LA  - en
ID  - 10_1017_fmp_2023_6
ER  - 
%0 Journal Article
%A Christopher Hacon
%A Jakub Witaszek
%T On the relative minimal model program for fourfolds in positive and mixed characteristic
%J Forum of Mathematics, Pi
%D 2023
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.6/
%R 10.1017/fmp.2023.6
%G en
%F 10_1017_fmp_2023_6
Christopher Hacon; Jakub Witaszek. On the relative minimal model program for fourfolds in positive and mixed characteristic. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.6

[ABL22] Arvidsson, E., Bernasconi, F. and Justin, L., ‘On the Kawamata–Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic’, Compos. Math. 158(4) (2022), 750–763. MR4438290.Google Scholar | DOI

[AHK07] Alexeev, V., Hacon, C. and Kawamata, Y., ‘Termination of (many) 4-dimensional log flips’, Invent. Math. 168(2) (2007), 433–448.Google Scholar | DOI

[BBE07] Berthelot, P., Bloch, S. and Esnault, H., ‘On Witt vector cohomology for singular varieties’, Compos. Math. 143(2) (2007), 363–392, MR2309991.Google Scholar | DOI

[BCHM10] Birkar, C., Cascini, P., Hacon, C. and Mckernan, J., ‘Existence of minimal models for varieties of log general type’, J. Amer. Math. Soc. 23(2) (2010), 405–468.Google Scholar | DOI

[Ber21] Bernasconi, F., ‘On the base point free theorem for klt threefolds in large characteristic’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22(2) (2021), 583–600. MR4288666.Google Scholar

[Bir16] Birkar, C., ‘Existence of flips and minimal models for 3-folds in char ’, Ann. Sci. Éc. Norm. Supér. (4) 49(1) (2016), 169–212.Google Scholar | DOI

[BMPSTWW20] Bhatt, B., Ma, L., Patakfalvi, Z., Schwede, K., Tucker, K., Waldron, J. and Witaszek, J., ‘Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic’, Preprint, 2020, .Google Scholar | arXiv

[BW17] Birkar, C. and Waldron, J., ‘Existence of Mori fibre spaces for 3-folds in ’, Adv. Math. 313 (2017), 62–101.Google Scholar | DOI

[CL16] Chiarellotto, B. and Lazda, C., ‘Combinatorial degenerations of surfaces and Calabi–Yau threefolds’, Algebra Number Theory 10(10) (2016), 2235–2266.Google Scholar | DOI

[CLL22] Chiarellotto, B., Lazda, C. and Liedtke, C., ‘Good reduction of K3 surfaces in equicharacteristic ’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23(1) (2022), 483–500. MR4407199.Google Scholar

[Cor07] Corti, A. (ed.), ‘Flips for 3-Folds and 4-Folds’, Oxford Lecture Series in Mathematics and Its Applications, vol. 35 (Oxford University Press, Oxford, 2007).Google Scholar

[CR12] Chatzistamatiou, A. and Rülling, K., ‘Hodge–Witt cohomology and Witt-rational singularities’, Doc. Math. 17 (2012), 663–781.Google Scholar | DOI

[CSK20] Carvajal-Rojas, J., Stäbler, A. and Kollár, J., ‘On the local étale fundamental group of klt threefold singularities’, Preprint, 2020, .Google Scholar | arXiv

[CT20] Cascini, P. and Tanaka, H., ‘Relative semi-ampleness in positive characteristic’, Proc. Lond. Math. Soc. (3) 121(3) (2020), 617–655. MR4100119.Google Scholar | DOI

[CTX15] Cascini, P., Tanaka, H. and Xu, C., ‘On base point freeness in positive characteristic’, Ann. Sci. Ecole Norm. Sup. 48(5) (2015), 1239–1272.Google Scholar | DOI

[Das15] Das, O., ‘On strongly -regular inversion of adjunction’, J. Algebra 434 (2015), 207–226.Google Scholar | DOI

[dFH11] De Fernex, T. and Hacon, C. D., ‘Deformations of canonical pairs and Fano varieties’, J. Reine Angew. Math. 651 (2011), 97–126. MR2774312.Google Scholar

[DW19a] Das, O. and Waldron, J., ‘On the abundance problem for 3-folds in characteristic 7’, Math. Z. 292(3–4) (2019), 937–946, MR3980278.5$+7’,+Math.+Z.+292(3–4)+(2019),+937–946,+MR3980278.>Google Scholar | DOI

[DW19b] Das, O. and Waldron, J., ‘On the log minimal model program for 3-folds over imperfect fields of characteristic ’, Preprint, 2019, .5$+’,+Preprint,+2019,+arXiv:1911.04394.>Google Scholar | arXiv

[Esn03] Esnault, H., ‘Varieties over a finite field with trivial Chow group of 0-cycles have a rational point’, Invent. Math. 151(1) (2003), 187–191.Google Scholar | DOI

[Fuj07] Fujino, O., ‘Special Termination and Reduction to pl Flips’, Flips for 3-Folds and 4-Folds (2007), 63–75.Google Scholar | DOI

[GNH19] Gongyo, Y., Nakamura, Y. and Hiromu, T., ‘Rational points on log Fano threefolds over a finite field’, J. Eur. Math. Soc. (JEMS) 21(12) (2019), 3759–3795. MR4022715.Google Scholar | DOI

[Gro62] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 4 (Société Mathématique de France, Paris, 2005). Séminaire de Géométrie Algébrique du Bois Marie, 1962, Augmenté d’un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud], With a preface and edited by Yves Laszlo, Revised reprint of the 1968 French original.Google Scholar

[Gro67] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV’, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361. MR238860.Google Scholar

[Har10] Hartshorne, R., Deformation Theory, Graduate Texts in Mathematics, vol. 257, (Springer, New York, 2010).Google Scholar

[Har77] Hartshorne, R., Algebraic Geometry (Springer-Verlag, New York, 1977).Google Scholar | DOI

[HMK10] Hacon, C. D. and Mckernan, J., ‘Existence of minimal models for varieties of log general type. II’, J. Amer. Math. Soc. 23(2) (2010), 469–490.Google Scholar | DOI

[HNT20] Hashizume, K., Nakamura, Y. and Tanaka, H., ‘Minimal model program for log canonical threefolds in positive characteristic’, Math. Res. Lett. 27(4) (2020), 1003–1054. MR4216577.Google Scholar | DOI

[HW19] Hacon, C. and Witaszek, J., ‘On the rationality of Kawamata log terminal singularities in positive characteristic’, Algebr. Geom. 6(5) (2019), 516–529. MR4009171.Google Scholar

[HW21] Hacon, C. and Witaszek, J., ‘On the relative minimal model program for threefolds in low characteristics’, Peking Math J. (2021).Google Scholar

[HW22] Hacon, C. and Witaszek, J., ‘The minimal model program for threefolds in characteristic 5’, Duke Mathematical Journal (2022), 1–39.Google Scholar

[HX15] Hacon, C. and Xu, C., ‘On the three dimensional minimal model program in positive characteristic’, J. Amer. Math. Soc. 28(3) (2015), 711–744.Google Scholar | DOI

[JS12] Jannsen, U. and Saito, S., ‘Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory’, J. Algebraic Geom. 21(4) (2012), 683–705.Google Scholar | DOI

[Kaw08] Kawamata, Y., ‘Flops connect minimal models’, Publ. RIMS, Kyoto Univ. 44 (2008), 419–423.Google Scholar | DOI

[Kaw94] Kawamata, Y., ‘Semistable minimal models of threefolds in positive or mixed characteristic’, J. Algebraic Geom. 3(3) (1994), 463–491.Google Scholar

[kee99] Keel, S., ‘Basepoint freeness for nef and big line bundles in positive characteristic’, Ann. of Math. (2) 149(1) (1999), 253–286.Google Scholar | DOI

[KM92] Kollár, J. and Mori, S., ‘Classification of three-dimensional flips’, J. Amer. Math. Soc. 5(3) (1992), 533–703.Google Scholar | DOI

[KM98] Kollár, J. and Mori, S., ‘Birational Geometry of Algebraic Varieties’, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).Google Scholar

[KMM87] Kawamata, Y., Matsuda, K. and Matsuki, K., ‘Introduction to the minimal model problem’, in Algebraic Geometry (Sendai, 1985, 1987), 283–360. MR946243.Google Scholar

[Kol13] Kollár, J., ‘Singularities of the Minimal Model Program’, Cambridge Tracts in Mathematics, vol. 200 (Cambridge University Press, Cambridge, 2013).Google Scholar

[Kol16] Kollár, J., ‘Variants of normality for Noetherian schemes’, Pure Appl. Math. Q. 12 (1) (2016), 1–31.Google Scholar | DOI

[Kol97] Kollár, J., ‘Quotient spaces modulo algebraic groups’, Ann. of Math. (2) 145(1) (1997), 33–79. MR1432036.Google Scholar | DOI

[LM18] Liedtke, C. and Yuya, M., ‘Good reduction of K3 surfaces’, Compos. Math. 154 (1) (2018), 1–35.Google Scholar | DOI

[LS14] Liedtke, C. and Matthew, S., ‘On the birational nature of lifting’, Adv. Math. 254 (2014), 118–137.Google Scholar | DOI

[Mau14] Maulik, D., ‘Supersingular K3 surfaces for large primes’, Duke Math. J. 163(13) (2014), 2357–2425. With an appendix by Andrew Snowden.Google Scholar | DOI

[MSTWW22] Ma, L., Schwede, K., Tucker, K., Waldron, J. and Witaszek, J., ‘An analog of adjoint ideals and plt singularities in mixed characteristic’, J. Algebraic Geom. 31 (2022), 497–559.Google Scholar | DOI

[NT20] Nakamura, Y. and Tanaka, H., ‘A Witt Nadel vanishing theorem for threefolds’, Compos. Math. 156(3) (2020), 435–475.Google Scholar | DOI

[NX16] Nicaise, J. and Xu, C., ‘The essential skeleton of a degeneration of algebraic varieties’, Amer. J. Math. 138(6) (2016), 1645–1667. MR3595497.Google Scholar | DOI

[Pop00] Popescu, D., ‘Artin approximation’, in Handbook of Algebra, vol. 2 (Elsevier, The Netherlands, 2000), 321–356.Google Scholar

[Sch14] Schwede, K., ‘A canonical linear system associated to adjoint divisors in characteristic ’, J. Reine Angew. Math. 696 (2014), 69–87.0$+’,+J.+Reine+Angew.+Math.+696+(2014),+69–87.>Google Scholar

[SS10] Schwede, K. and Smith, K. E., ‘Globally -regular and log Fano varieties’, Adv. Math. 224(3) (2010), 863–894.Google Scholar | DOI

[ST18] Sato, K. and Takagi, S., ‘General hyperplane sections of threefolds in positive characteristic’, Journal of the Institute of Mathematics of Jussieu (2018), 1–15.Google Scholar

[Sta14] The Stacks Project Authors, Stacks Project, 2014.Google Scholar

[SZ13] Schwede, K. and Zhang, W., ‘Bertini theorems for -singularities’, Proc. Lond. Math. Soc. (3) 107(4) (2013), 851–874. MR3108833.Google Scholar | DOI

[Tan18] Tanaka, H., ‘Minimal model program for excellent surfaces’, Ann. Inst. Fourier (Grenoble) 68(1) (2018), 345–376.Google Scholar | DOI

[TY20] Takamatsu, T. and Yoshikawa, S., ‘Minimal model program for semi-stable threefolds in mixed characteristic’, Preprint, 2020, .Google Scholar | arXiv

[Wal18] Waldron, J., ‘The LMMP for log canonical 3-folds in characteristic ’, Nagoya Math. J. 230 (2018), 48–71.5$+’,+Nagoya+Math.+J.+230+(2018),+48–71.>Google Scholar | DOI

[Wit21a] Witaszek, J., ‘On the canonical bundle formula and log abundance in positive characteristic’, Math. Ann. 381(3–4) (2021), 1309–1344. MR4333416.Google Scholar | DOI

[Wit21b] Witaszek, J., ‘Relative semiampleness in mixed characteristic’, Preprint, 2021, .Google Scholar | arXiv

[Wit22] Witaszek, J., ‘Keel’s base point free theorem and quotients in mixed characteristic’, Ann. of Math. (2) 195(2) (2022), 655–705.Google Scholar | DOI

[XX21a] Xie, L. and Xue, Q., ‘On the termination of the MMP for semi-stable fourfolds in mixed characteristic’, Preprint, 2021, .Google Scholar | arXiv

[XZ19] Xu, C. and Zhang, L., ‘Nonvanishing for 3-folds in characteristic ’, Duke Math. J. 168(7) (2019), 1269–1301. MR3953434.5$+’,+Duke+Math.+J.+168(7)+(2019),+1269–1301.+MR3953434.>Google Scholar | DOI

[Zda17] Zdanowicz, M., ‘Liftability of singularities and their Frobenius morphism modulo ’, Int. Math. Res. Not. IMRN (2017).Google Scholar | DOI

Cité par Sources :