On the log discrepancies in toric Mori contractions
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- by Valery Alexeev and Alexander Borisov PDF
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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 $\epsilon$ there is a positive $\delta$ such that if $X$ is $\epsilon$-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.References
- Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810. MR 1298994, DOI 10.1142/S0129167X94000395
- Florin Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), no. 2, 385–403. MR 2134273, DOI 10.1112/S0010437X04001071
- C. Birkar, Singularities on the base of a Fano type fibration, arXiv:1210.2658.
- 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, DOI 10.1070/SM1993v075n01ABEH003385
- Osamu Fujino, Applications of Kawamata’s positivity theorem, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 6, 75–79. MR 1712648
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Douglas Hensley, Lattice vertex polytopes with interior lattice points, Pacific J. Math. 105 (1983), no. 1, 183–191. MR 688412
- 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, DOI 10.1090/conm/207/02721
- Yujiro Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), no. 5, 893–899. MR 1646046
- 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
- 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, DOI 10.2969/aspm/01010283
- 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, DOI 10.4153/CJM-1991-058-4
- Shigefumi Mori and Yuri Prokhorov, On $\Bbb Q$-conic bundles, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 315–369. MR 2426350, DOI 10.2977/prims/1210167329
- Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
Additional Information
- Valery Alexeev
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30605
- MR Author ID: 317826
- Email: valery@math.uga.edu
- Alexander Borisov
- Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- Email: borisov@pitt.edu
- Received by editor(s): November 16, 2012
- Published electronically: July 3, 2014
- Communicated by: Ken Ono
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3251710