Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T13:38:47.639Z Has data issue: false hasContentIssue false
  • English
  • Français

On the semiampleness of the positive part of CKM Zariski decompositions

Published online by Cambridge University Press:  19 September 2013

SALVATORE CACCIOLA
SALVATORE CACCIOLA*
Affiliation:
Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146, Rome, Italy. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 1 , January 2014 , pp. 1 - 23
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AHK07]Alexeev, V., Hacon, C. and Kawamata, Y.Termination of (many) 4-dimensional log flips. Invent. Math. 168 (2007), no. 2, 433–448.CrossRefGoogle Scholar
[Amb01]Ambro, F.Quasi-log varieties. Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220–239; translation in Proc. Steklov Inst. Math. (2003), no. 1 (240), 214–233.Google Scholar
[Amb05]Ambro, F.A semiampleness criterion. J. Math. Sci. Univ. Tokyo 12 (2005), 445–466.Google Scholar
[BBP13]Boucksom, S., Broustet, A. and Pacienza, G.Uniruledness of stable base loci of adjoint linear Systems with and without Mori theory. To appear in Math. Z. DOI 10.1007/s00209-013-1144-y.Google Scholar
[BCHM10]Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J.Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2) (2010), 405–468.CrossRefGoogle Scholar
[BH13]Birkar, C. and Hu, Z. Log canonical pairs with good augmented base loci. arXiv: math.AG/1305.3569.Google Scholar
[ELMNP06]Ein, L., Lazarsfeld, R., Mustată, M., Nakamaye, M. and Popa, M.Asymptotic invariants of base loci. Ann. Inst. Fourier, Grenoble 56 (2006), p.1701, 1734.CrossRefGoogle Scholar
[Fuj79]Fujita, T.On Zariski problem. Proc. Japan Acad. Ser. A 55 (1979), 106–110.Google Scholar
[Fuj00]Fujino, O.Abundance theorem for semi log canonical threeefolds. Duke Math. J. 102 (2000), no. 3, 513–532.CrossRefGoogle Scholar
[Fuj07]Fujino, O.What is log terminal? Flips for 3-folds and 4-folds. Oxford Lecture Series in Mathematics and its Applications, vol. 35 (Oxford University Press, 2007), 49–62.CrossRefGoogle Scholar
[Fuj09]Fujino, O. Introduction to the log minimal model program for log canonical pairs. arXiv: math. AG/0907.1506v1.Google Scholar
[Fuj11]Fujino, O.Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789.CrossRefGoogle Scholar
[Fuj12]Fujino, O. Basepoint-free theorems: saturation, b-divisors, and canonical formula. Algebra Number Theory 6 (2012), no. 4, 797–823.CrossRefGoogle Scholar
[Gon12]Gongyo, Y. On weak Fano varieties with log canonical singularities. J. Reine. Angew. Math. 665 (2012), 237–252.Google Scholar
[Kaw87]Kawamata, Y.The Zariski decomposition of log-canonical divisors. Algebraic geometry. (Bowdoin, 1985) (Brunswick, Maine, 1985), 425–433, Proc. Sympos. Pure Math., 46, Part 1 (Amer. Math. Soc., Providence, RI, 1987).Google Scholar
[Kaw09]Kawamata, Y.Finite generation of a canonical ring. Current Developments in Mathematics (2007), 43–76 (Int. Press, Somerville, MA, 2009).CrossRefGoogle Scholar
[KM00]Kollár, J. and Mori, S. Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics 134.Google Scholar
[KMM85]Kawamata, Y., Matsuda, K. and Matsuki, K.Introduction to the minimal model problem, Algebraic geometry Sendai (1985), 283–360. Adv. Stud. Pure Math. 10 (North-Holland, Amsterdam, 1987).Google Scholar
[Kol05]Kollár, J.Resolution of Singularities - Seattle Lecture. arXiv:math.AG/0508332.Google Scholar
[Laz04]Lazarsfeld, R.Positivity in algebraic geometry, I–II. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 48–49 (Springer-Verlag, Berlin, 2004).Google Scholar
[Laz09]Lazić, V. Adjoint rings are finitely generated. arXiv: math. AG/0905.2707v1.Google Scholar
[Loh11]Lohmann, D. Families of canonically polarized manifolds over log Fano varieties. arXiv: math. AG/1107.4545.Google Scholar
[Mor87]Mori, S.Classification of higher-dimensional varieties. Algebraic geometry (Bowdoin, 1985) (Brunswick, Maine, 1985), 269-332, Proc. Sympos. Pure Math., 46, Part 1 (Amer. Math. Soc., Providence, RI, 1987).CrossRefGoogle Scholar
[Nak04]Nakayama, N.Zariski-decomposition and abundance. MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo (2004).Google Scholar
[Rei93]Reid, M.Commentary by M. Reid (chapter 10 of Shokurov's paper “3-fold log-flips”). Russian Acad. Sci. Izv. Math. 40 (1993), 195–200.Google Scholar
[Sza95]Szabó, E.Divisorial log terminal singularities. J. Math. Sci. Univ. Tokyo, 1 (1995) 631–639.Google Scholar