Abstract
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global \(F\)-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic.
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I. Aberbach and T. Polstra, Local cohomology bounds and the weak implies strong conjecture in dimension 4, J. Algebra, 605 (2022), 37–57, 4418959.
V. Alexeev, Boundedness and \(K^{2}\) for log surfaces, Int. J. Math., 5 (1994), 779–810, 1298994 (95k:14048).
V. Alexeev, C. Hacon and Y. Kawamata, Termination of (many) 4-dimensional log flips, Invent. Math., 168 (2007), 433–448.
Y. André, La conjecture du facteur direct, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 71–93, 3814651.
Y. André, Weak functoriality of Cohen-Macaulay algebras, J. Am. Math. Soc., 33 (2020), 363–380, 4073864.
M. Artin, Algebraization of formal moduli. II. Existence of modifications, Ann. Math. (2), 91 (1970), 88–135, MR0260747 (41 #5370).
M. Artin, On the joins of Hensel rings, Adv. Math., 7 (1971), 282–296, 289501.
F. Bernasconi, On the base point free theorem for klt threefolds in large characteristic, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 22 (2021), 583–600, 4288666.
F. Bernasconi and J. Kollár, Vanishing theorems for three-folds in characteristic \(p > 5\), Int. Math. Res. Not., 2023 (2023), 2846–2866.
B. Bhatt, Derived splinters in positive characteristic, Compos. Math., 148 (2012), 1757–1786, 2999303.
B. Bhatt, On the direct summand conjecture and its derived variant, Invent. Math., 212 (2018), 297–317, 3787829.
B. Bhatt, Cohen-Macaulayness of absolute integral closures, arXiv:2008.08070.
B. Bhatt and J. Lurie, A \(p\)-adic Riemann-Hilbert functor: \(\mathbf{Z}/p^{n}\)-coefficients, in preparation.
C. Birkar, Existence of flips and minimal models for 3-folds in char \(p\), Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 169–212.
C. Birkar and J. Waldron, Existence of Mori fibre spaces for 3-folds in \({{\mathrm{char}}}\,p\), Adv. Math., 313 (2017), 62–101.
C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Am. Math. Soc., 23 (2010), 405–468, 2601039 (2011f:14023).
M. Blickle, K. Schwede and K. Tucker, \(F\)-singularities via alterations, Am. J. Math., 137 (2015), 61–109, 3318087.
N. Bourbaki, Commutative Algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer, Berlin, 1998, Translated from the French, Reprint of the 1989 English translation. MR1727221 (2001g:13001).
N. Bourbaki, General Topology. Chapters 5–10, Elements of Mathematics (Berlin), Springer, Berlin, 1998, Translated from the French, Reprint of the 1989 English translation.
M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998, MR1613627 (99h:13020).
W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993, MR1251956 (95h:13020).
J. Carvajal-Rojas, K. Schwede and K. Tucker, Fundamental groups of \({F}\)-regular singularities via \({F}\)-signature, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 993–1016.
J. Carvajal-Rojas, L. Ma, T. Polstra, K. Schwede and K. Tucker, Covers of rational double points in mixed characteristic, J. Singul., 23 (2021), 127–150, 4292622.
P. Cascini and H. Tanaka, Relative semiampleness in positive characteristic, Proc. Lond. Math. Soc. (3), 121 (2020), 617–655.
P. Cascini, C. Hacon, M. Mustaţă and K. Schwede, On the numerical dimension of pseudo-effective divisors in positive characteristic, Am. J. Math., 136 (2014), 1609–1628, 3282982.
P. Cascini, H. Tanaka and C. Xu, On base point freeness in positive characteristic, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1239–1272.
P. Cascini, H. Tanaka and J. Witaszek, On log del Pezzo surfaces in large characteristic, Compos. Math., 153 (2017), 820–850, 3621617.
A. Chiecchio, F. Enescu, L. E. Miller and K. Schwede, Test ideals in rings with finitely generated anti-canonical algebras, J. Inst. Math. Jussieu, 17 (2018), 171–206, 3742559.
B. Conrad, Grothendieck Duality and Base Change, Lecture Notes in Mathematics, vol. 1750, Springer, Berlin, 2000, MR1804902 (2002d:14025).
B. Conrad, The Keel-Mori theorem via stacks, 2005, http://math.stanford.edu/~conrad/papers/coarsespace.pdf.
A. Corti, Flips for 3-Folds and 4-Folds, Oxford Lecture Series in Mathematics and Its Applications, vol. 35, Oxford University Press, Oxford, 2007, 2352762 (2008j:14031).
V. Cossart and O. Piltant, Resolution of singularities of arithmetical threefolds, J. Algebra, 529 (2019), 268–535, 3942183.
V. Cossart, U. Jannsen and S. Saito, Desingularization: Invariants and Strategy, Lecture Notes in Mathematics, vol. 2270, Springer, Berlin, 2020.
O. Das, On strongly \({F}\)-regular inversion of adjunction, J. Algebra, 434 (2015), 207–226.
O. Das and J. Waldron, On the log minimal model program for threefolds over imperfect fields of characteristic \(p>5\), J. Lond. Math. Soc. (2022). https://doi.org/10.1112/jlms.12677.
R. Datta and T. Murayama, Tate algebras and Frobenius non-splitting of excellent regular rings, J. Eur. Math. Soc. (2020), in press. arXiv:2003.13714.
R. Datta and K. Tucker, Openness of splinter loci in prime characteristic, arXiv:2103.10525, 2021.
P. Deligne and N. Katz (eds.), Groupes de monodromie en géométrie algébrique. II. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, vol. 340, Springer, Berlin, 1973, 0354657 (50 #7135).
J.-P. Demailly, A numerical criterion for very ample line bundles, J. Differ. Geom., 37 (1993), 323–374, 1205448 (94d:14007).
D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995, With a view toward algebraic geometry, 1322960 (97a:13001).
S. Ejiri, When is the Albanese morphism an algebraic fiber space in positive characteristic? Manuscr. Math., 160 (2019), 239–264, 3983395.
T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Publ. Math. Inst. Hautes Études Sci., 67 (1988), 97–144, 972344 (89k:14069).
H. Flenner, Die Sätze von Bertini für lokale Ringe, Math. Ann., 229 (1977), 97–111, 0460317.
H. Flenner, L. O’Carroll and W. Vogel, Joins and Intersections, Springer Monographs in Mathematics, Springer, Berlin, 1999, 1724388.
H.-B. r. Foxby and S. Iyengar, Depth and amplitude for unbounded complexes, in Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math., vol. 331, pp. 119–137, Am. Math. Soc., Providence, 2003, 2013162.
O. Fujino, Special termination and reduction to pl flips, in Flips for 3-Folds and 4-Folds, Oxford Lecture Ser. Math. Appl., vol. 35, pp. 63–75, Oxford Univ. Press, Oxford, 2007.
O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci., 47 (2011), 727–789, 2832805.
O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci., 48 (2012), 339–371.
O. Fujino and K. Miyamoto, Nakai–Moishezon ampleness criterion for real line bundles, Math. Ann., 385 (2022), 1–12. https://doi.org/10.1007/s00208-021-02354-9. arXiv:2101.00806.
O. Gabber, Notes on some \(t\)-structures, in Geometric Aspects of Dwork Theory. Vol. I, II, pp. 711–734, de Gruyter, Berlin, 2004.
M. Ghosh and A. Krishna, Bertini theorems revisited, arXiv:1912.09076.
Y. Gongyo, Z. Li, Z. Patakfalvi, K. Schwede, H. Tanaka and R. Zong, On rational connectedness of globally \(F\)-regular threefolds, Adv. Math., 280 (2015), 47–78.
Y. Gongyo, Y. Nakamura and H. Tanaka, Rational points on log Fano threefolds over a finite field, J. Eur. Math. Soc., 21 (2019), 3759–3795.
A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci., 8 (1961), 222, MR0217084 (36 #177b).
C. D. Hacon, Singularities of pluri-theta divisors in Char \(p>0\), in Algebraic Geometry in East Asia—Taipei 2011, Adv. Stud. Pure Math., vol. 65, pp. 117–122, Math. Soc. Japan, Tokyo, 2015, 3380778.
C. D. Hacon and S. J. Kovács, On the boundedness of slc surfaces of general type, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 19 (2019), 191–215, 3923845.
C. D. Hacon and Z. Patakfalvi, Generic vanishing in characteristic \(p>0\) and the characterization of ordinary abelian varieties, Am. J. Math., 138 (2016), 963–998, 3538148.
C. Hacon and J. Witaszek, The minimal model program for threefolds in characteristic 5, Duke Math. J., 171 (2022), 2193–2231, 4484207.
C. Hacon and J. Witaszek, On the relative minimal model program for threefolds in low characteristics, Peking Math. J., 5 (2022), 365–382, 4492657.
C. Hacon and J. Witaszek, On the relative minimal model program for fourfolds in positive and mixed characteristic, Forum Math. Pi, 11 (2023), e10, 4565409.
C. D. Hacon and C. Xu, On the three dimensional minimal model program in positive characteristic, J. Am. Math. Soc., 28 (2015), 711–744.
C. D. Hacon, Z. Patakfalvi and L. Zhang, Birational characterization of abelian varieties and ordinary abelian varieties in characteristic \(p>0\), Duke Math. J., 168 (2019), 1723–1736.
N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Am. J. Math., 120 (1998), 981–996, MR1646049 (99h:13005).
N. Hara and K.-I. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom., 11 (2002), 363–392, MR1874118 (2002k:13009).
N. Hara and K.-I. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Am. Math. Soc., 355 (2003), 3143–3174, (electronic). MR1974679 (2004i:13003).
R. Hartshorne, Residues and Duality. Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963/64, Lecture Notes in Mathematics, vol. 20, Springer, Berlin, 1966, With an Appendix by P. Deligne, MR0222093 (36 #5145).
R. Hartshorne, Local Cohomology. A Seminar Given by A. Grothendieck, Harvard University, Fall, Springer, Berlin, 1967, MR0224620 (37 #219).
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, New York, 1977, MR0463157 (57 #3116).
R. Hartshorne, Generalized divisors on Gorenstein schemes, in Proceedings of Conference on Algebraic Geometry and Ring Theory in Honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, pp. 287–339, 1994, MR1291023 (95k:14008).
R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic \(p\), Ann. Math. (2), 105 (1977), 45–79, MR0441962 (56 #353).
K. Hashizume, Y. Nakamura and H. Tanaka, Minimal model program for log canonical threefolds in positive characteristic, Math. Res. Lett., 27 (2020), 1003–1054.
M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J., 51 (1973), 25–43, 0349656 (50 #2149).
M. Hochster, Topics in the homological theory of modules over commutative rings, in Conference Board of the Mathematical Sciences, Am. Math. Soc., Providence, 1975, Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24–28, 1974, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24. 0371879 (51 #8096).
M. Hochster, in Solid Closure, Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, MA, 1992), Contemp. Math., vol. 159, pp. 103–172, Am. Math. Soc., Providence, 1994, 1266182.
M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Am. Math. Soc., 3 (1990), 31–116, MR1017784 (91g:13010).
M. Hochster and C. Huneke, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. Math. (2), 135 (1992), 53–89, 1147957 (92m:13023).
M. Hochster and C. Huneke, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math., 113 (1995), 45–117, 1332808.
C. Huneke and G. Lyubeznik, Absolute integral closure in positive characteristic, Adv. Math., 210 (2007), 498–504, 2303230 (2008d:13005).
E. Hyry and K. E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Am. J. Math., 125 (2003), 1349–1410, MR2018664 (2006c:13036).
L. Ji and J. Waldron, Structure of geometrically non-reduced varieties, Trans. Am. Math. Soc., in press. https://doi.org/10.1090/tran/8388.
Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79 (1985), 567–588.
Y. Kawamata, Semistable minimal models of threefolds in positive or mixed characteristic, J. Algebraic Geom., 3 (1994), 463–491.
Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, in Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math., vol. 10, pp. 283–360, North-Holland, Amsterdam, 1987, MR946243 (89e:14015).
S. Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. Math. (2), 149 (1999), 253–286, 1680559 (2000j:14011).
S. Keel and S. Mori, Quotients by groupoids, Ann. Math. (2), 145 (1997), 193–213, 1432041 (97m:14014).
D. S. Keeler, Fujita’s conjecture and Frobenius amplitude, Am. J. Math., 130 (2008), 1327–1336, 2450210 (2009i:14006).
S. L. Kleiman, Toward a numerical theory of ampleness, Ann. Math., 84 (1966), 293–344.
J. Kollár, Projectivity of complete moduli, J. Differ. Geom., 32 (1990), 235–268, 1064874 (92e:14008).
J. Kollár, Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013, With the collaboration of Sándor Kovács, 3057950.
J. Kollár, Relative MMP without \(\mathbf {Q}\)-factoriality, Electron. Res. Arch., 29 (2021), 3193–3203.
J. Kollár, Exercises in the birational geometry of algebraic varieties, arXiv:0809.2579.
J. Kollár, Hulls and husks, arXiv:0805.0576.
J. Kollár and S. 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, MR1658959 (2000b:14018).
J. Kollár and J. Witaszek, Resolution and alteration with ample exceptional, arXiv:2102.03162, 2021.
J. Kollár, et al., Flips and abundance for algebraic threefolds, in Second Summer Seminar on Algebraic Geometry (University of Utah, Salt Lake City, Utah, August 1991), Astérisque, vol. 211, Société Mathématique de France, Paris, 1992, MR1225842 (94f:14013).
S. Kovács, Rational singularities, arXiv:1703.02269.
R. Lazarsfeld, Positivity in algebraic geometry. I: classical setting: line bundles and linear series, in Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge [Results in Mathematics and Related Areas. 3rd Series] A Series of Modern Surveys in Mathematics, vol. 48, Springer, Berlin, 2004, MR2095471 (2005k:14001a).
S. Lichtenbaum, Curves over discrete valuation rings, Am. J. Math., 90 (1968), 380–405, 230724.
C. Liedtke, Algebraic surfaces of general type with small \(c_{1}^{2}\) in positive characteristic, Nagoya Math. J., 191 (2008), 111–134, 2451222 (2009i:14052).
J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci., 36 (1969), 195–279, MR0276239 (43 #1986).
J. Lipman, Desingularization of two-dimensional schemes, Ann. Math. (2), 107 (1978), 151–207, 0491722 (58 #10924).
Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002, 1917232 (2003g:14001).
G. Lyubeznik, On the vanishing of local cohomology in characteristic \(p>0\), Compos. Math., 142 (2006), 207–221.
L. Ma and K. Schwede, Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers, Invent. Math., 214 (2018), 913–955, 3867632.
L. Ma and K. Schwede, Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras, Duke Math. J., 170 (2021), 2815–2890, 4312190.
L. Ma, K. Schwede, K. Tucker, J. Waldron and J. Witaszek, An analogue of adjoint ideals and PLT singularities in mixed characteristic, J. Algebraic Geom., 31 (2022), 497–559, 4484548.
H. 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, MR1011461 (90i:13001).
V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. (2), 122 (1985), 27–40, MR799251 (86k:14038).
V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math., 1 (1997), 249–271, MR1491985 (99e:13009).
T. Murayama, Relative vanishing theorems for \(\mathbf {Q}\)-schemes, arXiv:2101.10397, 2021.
M. Mustaţǎ and K. Schwede, A Frobenius variant of Seshadri constants, Math. Ann., 358 (2014), 861–878.
Z. Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 991–1025, 3294622.
Zs. Patakfalvi, On the projectivity of the moduli space of stable surfaces in characteristic \(p >5\), arXiv:1710.03818.
Z. Patakfalvi and J. Waldron, Singularities of general fibers and the LMMP, Am. J. Math., 144 (2022), 505–540.
S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math., 79 (1985), 217–224, MR778124 (86j:14051).
M. Raynaud, Contre-exemple au “Vanishing Theorem” en caractéristique \(p>0\), in Tata Inst. Fund. Res. Studies in Math., vol. 8, pp. 273–278, Springer, Berlin, 1978, C. P. Ramanujam—a tribute, 541027 (81b:14011).
M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., 13 (1971), 1–89.
T. Saito, Log smooth extension of a family of curves and semi-stable reduction, J. Algebraic Geom., 13 (2004), 287–321, 2047700 (2005a:14034).
P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci., 116 (2012), 245–313.
K. Schwede, A canonical linear system associated to adjoint divisors in characteristic \(p > 0\), J. Reine Angew. Math., 696 (2014), 69–87.
K. Schwede and K. E. Smith, Globally \(F\)-regular and log Fano varieties, Adv. Math., 224 (2010), 863–894, 2628797 (2011e:14076).
K. Schwede and S. Takagi, Rational singularities associated to pairs, Mich. Math. J., 57 (2008), 625–658.
K. Schwede and K. Tucker, Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems, J. Math. Pures Appl. (9), 102 (2014), 891–929, 3271293.
K. Schwede, K. Tucker and W. Zhang, Test ideals via a single alteration and discreteness and rationality of \(F\)-jumping numbers, Math. Res. Lett., 19 (2012), 191–197, 2923185.
I. R. Shafarevich, Lectures on minimal models and birational transformations of two dimensional schemes, in Notes by C. P. Ramanujam, Lectures on Mathematics and Physics, vol. 37, Tata Institute of Fundamental Research, Bombay, 1966, 0217068.
K. Shimomoto and E. Tavanfar, On local rings without small Cohen-Macaulay algebras in mixed characteristic, arXiv:2109.12700.
V. V. Shokurov, Three-dimensional log perestroikas, Izv. Akad. Nauk SSSR, Ser. Mat., 56 (1992), 105–203, MR1162635 (93j:14012).
A. K. Singh, \(\mathbf{Q}\)-Gorenstein splinter rings of characteristic \(p\) are F-regular, Math. Proc. Camb. Philos. Soc., 127 (1999), 201–205, 1735920 (2000j:13006).
K. E. Smith, \(F\)-rational rings have rational singularities, Am. J. Math., 119 (1997), 159–180, MR1428062 (97k:13004).
K. E. Smith, Fujita’s freeness conjecture in terms of local cohomology, J. Algebraic Geom., 6 (1997), 417–429, MR1487221 (98m:14002).
K. E. Smith, Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Mich. Math. J., 48 (2000), 553–572, Dedicated to William Fulton on the occasion of his 60th birthday, MR1786505 (2001k:13007).
J. Starr, Bounding the number of critical points in a Lefschetz pencil, 2013, https://mathoverflow.net/q/165674 (Version: 2014-05-09).
S. Takagi, F-singularities of pairs and inversion of adjunction of arbitrary codimension, Invent. Math., 157 (2004), 123–146, MR2135186.
S. Takagi, An interpretation of multiplier ideals via tight closure, J. Algebraic Geom., 13 (2004), 393–415, MR2047704 (2005c:13002).
T. Takamatsu and S. Yoshikawa, Minimal model program for semi-stable threefolds in mixed characteristic, J. Algebr. Geom., in press. https://doi.org/10.1090/jag/813. arXiv:2012.07324.
H. Tanaka, Behavior of canonical divisors under purely inseparable base changes, J. Reine Angew. Math., 744 (2018), 237–264.
H. Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble), 68 (2018), 345–376, 3795482.
H. Tanaka, Abundance theorem for surfaces over imperfect fields, Math. Z., 295 (2020), 595–622.
H. Tanaka, Pathologies on Mori fibre spaces in positive characteristic, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 20 (2020), 1113–1134.
H. Tanaka, Private Communication, 2020.
M. Temkin, Absolute desingularization in characteristic zero, in Motivic Integration and Its Interactions with Model Theory and Non-Archimedean Geometry, Vol. II, London Math. Soc. Lecture Note Ser., vol. 384, pp. 213–250, Cambridge University Press, Cambridge, 2011, 2905858.
The Stacks Project Authors, Stacks Project.
M. Tomari and K. Watanabe, Filtered rings, filtered blowing-ups and normal two-dimensional singularities with “star-shaped” resolution, Publ. Res. Inst. Math. Sci., 25 (1989), 681–740, 1031224.
B. Totaro, The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen-Macaulay, J. Algebraic Geom., 28 (2019), 751–771, 3994312.
B. Totaro, Private Communication, 2021.
V. Trivedi, Erratum: “a local Bertini theorem in mixed characteristic” [Comm. Algebra 22 (1994), no. 3, 823–827; MR1261007 (94m:13002)], Commun. Algebra, 25 (1997), 1685–1686, 1444028.
T. Vijaylaxmi, A local Bertini theorem in mixed characteristic, Commun. Algebra, 22 (1994), 823–827, 1261007.
J. Waldron, The LMMP for log canonical 3-folds in char \(p\), Nagoya Math. J., 230 (2018), 48–71.
K. Watanabe, \(F\)-regular and \(F\)-pure normal graded rings, J. Pure Appl. Algebra, 71 (1991), 341–350.
J. Witaszek, Keel’s base point free theorem and quotients in mixed characteristic, Ann. Math. (2), 195 (2022), 655–705, 4387235.
J. Witaszek, Relative semiampleness in mixed characteristic, arXiv:2106.06088.
F. B. Wright, Semigroups in compact groups, Proc. Am. Math. Soc., 7 (1956), 309–311.
L. Xie and Q. Xue, On the termination of the mmp for semi-stable fourfolds in mixed characteristic, arXiv:2110.03115, 2021.
C. Xu, On the base-point-free theorem of 3-folds in positive characteristic, J. Inst. Math. Jussieu, 14 (2015), 577–588.
C. Xu and L. Zhang, Nonvanishing for 3-folds in characteristic \(p>5\), Duke Math. J., 168 (2019), 1269–1301.
Y. Zhang, Pluri-canonical maps of varieties of maximal Albanese dimension in positive characteristic, J. Algebra, 409 (2014), 11–25, 3198833.
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Bhatt, B., Ma, L., Patakfalvi, Z. et al. Globally \(\pmb{+}\)-regular varieties and the minimal model program for threefolds in mixed characteristic. Publ.math.IHES 138, 69–227 (2023). https://doi.org/10.1007/s10240-023-00140-8
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DOI: https://doi.org/10.1007/s10240-023-00140-8