On the three dimensional minimal model program in positive characteristic

Hacon, Christopher; Xu, Chenyang

doi:10.1090/S0894-0347-2014-00809-2

On the three dimensional minimal model program in positive characteristic
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by Christopher D. Hacon and Chenyang Xu

J. Amer. Math. Soc. 28 (2015), 711-744

DOI: https://doi.org/10.1090/S0894-0347-2014-00809-2

Published electronically: June 4, 2014

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Abstract:

Let $f:(X,B)\to Z$ be a threefold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{ B\}$ are in the standard set $\{ 1-\frac 1n|n\in \mathbb N\}$, then the flip of $f$ exists. As a consequence, we prove the existence of minimal models for any projective ${\mathbb Q}$-factorial terminal variety $X$ with pseudo-effective canonical divisor $K_X$.

References

S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523, DOI 10.1007/978-3-662-03580-1
Caucher Birkar, Paolo Cascini, Christopher Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468., DOI 10.1090/S0894-0347-09-00649-3
Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (2008), no. 3, 1051–1082. MR 2427629, DOI 10.1016/j.jalgebra.2008.03.032
Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321 (2009), no. 7, 1836–1976. MR 2494751, DOI 10.1016/j.jalgebra.2008.11.030
Alessio Corti, 3-fold flips after Shokurov, Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl., vol. 35, Oxford Univ. Press, Oxford, 2007, pp. 18–48. MR 2359340, DOI 10.1093/acprof:oso/9780198570615.003.0002
Steven Dale Cutkosky, Resolution of singularities, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004. MR 2058431, DOI 10.1090/gsm/063
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
Osamu Fujino, Karl Schwede, and Shunsuke Takagi, Supplements to non-lc ideal sheaves, Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, B24, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 1–46. MR 2809647
Nobuo Hara, Classification of two-dimensional $F$-regular and $F$-pure singularities, Adv. Math. 133 (1998), no. 1, 33–53. MR 1492785, DOI 10.1006/aima.1997.1682
Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, DOI 10.1090/S0894-0347-1990-1017784-6
Jen-Chieh Hsiao, Karl Schwede, and Wenliang Zhang, Cartier modules on toric varieties, Trans. Amer. Math. Soc. 366 (2014), no. 4, 1773–1795. MR 3152712, DOI 10.1090/S0002-9947-2013-05856-4
Yujiro Kawamata, Semistable minimal models of threefolds in positive or mixed characteristic, J. Algebraic Geom. 3 (1994), no. 3, 463–491. MR 1269717
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
Dennis S. Keeler, Ample filters of invertible sheaves, J. Algebra 259 (2003), no. 1, 243–283. MR 1953719, DOI 10.1016/S0021-8693(02)00557-4
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
K. Kollár and S. Kovács, Birational geometry of log surfaces, available at https://web.math.princeton.edu/~kollar/.
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
Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269–331. MR 927961, DOI 10.1090/pspum/046.1/927961
V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40. MR 799251, DOI 10.2307/1971368
Mircea Mustaţă and Karl Schwede, A Frobenius variant of Seshadri constants, Math. Ann. 358 (2014), no. 3-4, 861–878. MR 3175143, DOI 10.1007/s00208-013-0976-4
Z. Patakfalvi, Semi-positivity in positive characteristics, Annales scientifiques de l’École Normale Supérieure. to appear.
Yuri G. Prokhorov, Lectures on complements on log surfaces, MSJ Memoirs, vol. 10, Mathematical Society of Japan, Tokyo, 2001. MR 1830440
Karl Schwede, $F$-adjunction, Algebra Number Theory 3 (2009), no. 8, 907–950. MR 2587408, DOI 10.2140/ant.2009.3.907
Karl Schwede, Centers of $F$-purity, Math. Z. 265 (2010), no. 3, 687–714. MR 2644316, DOI 10.1007/s00209-009-0536-5
K. Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, Journal für die reine und angewandte Mathematik. to appear.
K. Schwede. Personal communication, 2013.
Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI 10.1016/j.aim.2009.12.020
K. Schwede and K. Tucker, On the behavior of test ideals under finite morphisms, J. Algebraic Geom. 23 (2014), 399–443., DOI 10.1090/S1056-3911-2013-00610-4
Karl Schwede and Kevin Tucker, A survey of test ideals, Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012, pp. 39–99. MR 2932591
V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, DOI 10.1070/IM1993v040n01ABEH001862
V. V. Shokurov, Complements on surfaces, J. Math. Sci. (New York) 102 (2000), no. 2, 3876–3932. Algebraic geometry, 10. MR 1794169, DOI 10.1007/BF02984106
H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J., available at arXiv:1201.5699.
H. Tanaka, X-method for klt surfaces in positive characteristic, J. Algebraic Geom., available at arXiv:1202.2497.
Keiichi Watanabe, $F$-regular and $F$-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), no. 2-3, 341–350. MR 1117644, DOI 10.1016/0022-4049(91)90157-W