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Minimal Models and Abundance for Positive Characteristic Log Surfaces

Published online by Cambridge University Press:  11 January 2016

Hiromu Tanaka
Hiromu Tanaka*
Affiliation:
Department of Mathematics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan, tanakahi@math.kobe-u.ac.jp
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Abstract

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We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

Keywords

14E3014J10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 216 , 2014 , pp. 1 - 70
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[A] Artin, M., Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 0146182.Google Scholar
[Bă1] Bădescu, L., Algebraic Surfaces, Springer, New York, 2001. MR 1805816. DOI 10.1007/978-1-4757-3512-3.CrossRefGoogle Scholar
[Bă2] Bădescu, L., “On some contractibility criteria of curves on surfaces” in Sitzungsberichte der Berliner Mathematischen Gesellschaft, Berliner Math. Gesellschaft, Berlin, 2001, 41–51. MR 2091877.Google Scholar
[Bi] Birkar, C., On existence of log minimal models, II, J. Reine Angew. Math. 658 (2011), 99–113. MR 2831514. DOI 10.1515/CRELLE.2011.062.Google Scholar
[BM1] Bombieri, E. and Mumford, D., “Enriques’ classification of surfaces in char p, II” in Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 23–42. MR 0491719.CrossRefGoogle Scholar
[BM2] Bombieri, E. and Mumford, D., Enriques’ classification of surfaces in char p, III, Invent. Math. 35 (1976), 197–232. MR 0491720.CrossRefGoogle Scholar
[CMM] Cascini, P., McKernan, J., and Mustaţă, M., The augmented base locus in positive characteristic, Proc. Edinb. Math. Soc. (2) 57 (2014), 79–87. MR 3165013. DOI 10.1017/S0013091513000916.Google Scholar
[FJ] Frey, G. and Jarden, M., Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. Lond. Math. Soc. (3) 28 (1974), 112–128. MR 0337997.CrossRefGoogle Scholar
[F1] Fujino, O., Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), 727–789. MR 2832805. DOI 10.2977/PRIMS/50.Google Scholar
[F2] Fujino, O., Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), 339–371. MR 2928144. DOI 10.2977/PRIMS/71.CrossRefGoogle Scholar
[FT] Fujino, O. and Tanaka, H., On log surfaces, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), 109–114. MR 2989060. DOI 10.3792/pjaa.88.109.Google Scholar
[Fu] Fujita, T., Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), 685–696. MR 0731524.Google Scholar
[H] Hartshorne, R., Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. MR 0463157.Google Scholar
[K] Kawamata, Y., Semistable minimal models of threefolds in positive or mixed characteristic, J. Algebraic Geom. 3 (1994), 463–491. MR 1269717.Google Scholar
[KMM] Kawamata, Y., Matsuda, K., and Matsuki, K., “Introduction to the minimal model problem” in Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 283–360. MR 0946243.Google Scholar
[Ke1] Keel, S., Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), 253–286. MR 1680559. DOI 10.2307/121025.Google Scholar
[Ke2] Keel, S., Polarized pushouts over finite fields, Comm. Algebra 31 (2003), 3955–3982. MR 2007391. DOI 10.1081/AGB-120022449.CrossRefGoogle Scholar
[Ko1] Kollár, J., Extremal rays on smooth threefolds, Ann. Sci. Éc. Norm. Supár. (4) 24 (1991), 339–361. MR 1100994.Google Scholar
[Ko2] Kollár, J., Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996. MR 1440180. DOI 10.1007/978-3-662-03276-3.Google Scholar
[KoK] Kollár, J. and Kovács, S., Birational geometry of log surfaces, preprint.Google Scholar
[KoM] Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge, 1998. MR 1658959. DOI 10.1017/CBO9780511662560.Google Scholar
[L] Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Etudes Sci. 36 (1969), 195–279. MR 0276239.Google Scholar
[M] Másek, V., Kodaira–Iitaka and numerical dimensions of algebraic surfaces over the algebraic closure of a finite field, Rev. Roumaine Math. Pures Appl. 38 (1993), 679–685. MR 1263211.Google Scholar
[Ma] Matsumura, H., Commutative Algebra, Benjamin, New York, 1970. MR 0266911.Google Scholar
[Mi] Miyanishi, M., Noncomplete Algebraic Surfaces, Lecture Notes in Math. 857, Springer, Berlin, 1981. MR 0635930.Google Scholar
[Mo1] Mori, S., Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), 593–606. MR 0554387. DOI 10.2307/1971241.CrossRefGoogle Scholar
[Mo2] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), 133–176. MR 0662120. DOI 10.2307/2007050.CrossRefGoogle Scholar
[Mu1] Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Etudes Sci. 9 (1961), 5–22. MR 0153682.Google Scholar
[Mu2] Mumford, D., “Enriques’ classification of surfaces in char p, I” in Global Analysis (Papers in Honor of Kodaira, K.), Univ. Tokyo Press, Tokyo, 1969, 325–339. MR 0254053.Google Scholar
[R] Raynaud, M., “Contre-exemple au ‘vanishing theorem’ en caractéristique p > 0” in C. P. Ramanujam—A Tribute, Tata Inst. Fund. Res. Stud. Math. 8, Springer, Berlin, 1978, 273–278. MR 0541027.Google Scholar
[S] Sakai, F., “Classification of normal surfaces” in Algebraic Geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Pt. 1, Amer. Math. Soc., Providence, 1987, 451–465. MR 0927967.Google Scholar
[T] Tanaka, H., The X-method for klt surfaces in positive characteristic, preprint, 2012.Google Scholar
[To] Totaro, B., Moving codimension-one subvarieties over finite fields, Amer. J. Math. 131 (2009), 1815–1833. MR 2567508. DOI 10.1353/ajm.0.0088.Google Scholar