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On the log discrepancies in toric Mori contractions


Authors: Valery Alexeev and Alexander Borisov
Journal: Proc. Amer. Math. Soc. 142 (2014), 3687-3694
MSC (2010): Primary 14E30, 14M25
DOI: https://doi.org/10.1090/S0002-9939-2014-12159-9
Published electronically: July 3, 2014
MathSciNet review: 3251710
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Abstract: It was conjectured by McKernan and Shokurov that for all Mori contractions from $ X$ to $ Y$ of given dimensions, for any positive $ \eps $ there is a positive $ \delta $ such that if $ X$ is $ \eps $-log terminal, then $ Y$ is $ \delta $-log terminal. We prove this conjecture in the toric case and discuss the dependence of $ \delta $ on $ \epsilon $, which seems mysterious.


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  • [Ale94] Valery Alexeev, Boundedness and $ K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779-810. MR 1298994 (95k:14048), https://doi.org/10.1142/S0129167X94000395
  • [Amb05] Florin Ambro, The moduli $ b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), no. 2, 385-403. MR 2134273 (2006d:14015), https://doi.org/10.1112/S0010437X04001071
  • [Bir12] C. Birkar, Singularities on the base of a Fano type fibration, arXiv:1210.2658.
  • [BB92] A. A. Borisov and L. A. Borisov, Singular toric Fano three-folds, Mat. Sb. 183 (1992), no. 2, 134-141 (Russian); English transl., Russian Acad. Sci. Sb. Math. 75 (1993), no. 1, 277-283. MR 1166957 (93i:14034), https://doi.org/10.1070/SM1993v075n01ABEH003385
  • [Fuj99] Osamu Fujino, Applications of Kawamata's positivity theorem, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 6, 75-79. MR 1712648 (2000f:14089)
  • [Ful93] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993. MR 1234037 (94g:14028)
  • [Hen83] Douglas Hensley, Lattice vertex polytopes with interior lattice points, Pacific J. Math. 105 (1983), no. 1, 183-191. MR 688412 (84c:52016)
  • [Kaw97] Yujiro Kawamata, Subadjunction of log canonical divisors for a subvariety of codimension $ 2$, Birational algebraic geometry (Baltimore, MD, 1996) Contemp. Math., vol. 207, Amer. Math. Soc., Providence, RI, 1997, pp. 79-88. MR 1462926 (99a:14024), https://doi.org/10.1090/conm/207/02721
  • [Kaw98] Yujiro Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), no. 5, 893-899. MR 1646046 (2000d:14020)
  • [KM98] János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, with the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. MR 1658959 (2000b:14018)
  • [KMM87] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283-360. MR 946243 (89e:14015)
  • [LZ91] Jeffrey C. Lagarias and Günter M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canad. J. Math. 43 (1991), no. 5, 1022-1035. MR 1138580 (92k:52032), https://doi.org/10.4153/CJM-1991-058-4
  • [MP08] Shigefumi Mori and Yuri Prokhorov, On $ \mathbb{Q}$-conic bundles, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 315-369. MR 2426350 (2009e:14062), https://doi.org/10.2977/prims/1210167329
  • [Oda88] Tadao Oda, Convex bodies and algebraic geometry, An introduction to the theory of toric varieties. Translated from the Japanese. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. MR 922894 (88m:14038)

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Additional Information

Valery Alexeev
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30605
Email: valery@math.uga.edu

Alexander Borisov
Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: borisov@pitt.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12159-9
Received by editor(s): November 16, 2012
Published electronically: July 3, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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