Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T05:53:20.931Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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