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Published online by Cambridge University Press:  29 August 2018

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan email
Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 8-19-1, Nanakuma, Jonan-ku, Fukuoka 814-0180, Japan email


We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita’s freeness and very ampleness for toric varieties.

© 2018 Foundation Nagoya Mathematical Journal

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Cox, D. A., Little, J. B. and Schenck, H. K., Toric Varieties, Graduate Studies in Mathematics 124, American Mathematical Society, Providence, RI, 2011.10.1090/gsm/124Google Scholar
Fujino, O., Notes on toric varieties from Mori theoretic viewpoint, Tohoku Math. J. (2) 55(4) (2003), 551564.10.2748/tmj/1113247130Google Scholar
Fujino, O., On the Kleiman–Mori cone, Proc. Japan Acad. Ser. A Math. Sci. 81(5) (2005), 8084.10.3792/pjaa.81.80Google Scholar
Fujino, O., Equivariant completions of toric contraction morphisms, Tohoku Math. J. (2) 58(3) (2006), 303321.10.2748/tmj/1163775132Google Scholar
Fujino, O., Toric varieties whose canonical divisors are divisible by their dimensions, Osaka J. Math. 43(2) (2006), 275281.Google Scholar
Fujino, O., Multiplication maps and vanishing theorems for toric varieties, Math. Z. 257(3) (2007), 631641.10.1007/s00209-007-0140-5Google Scholar
Fujino, O., “Vanishing theorems for toric polyhedra”, in Higher Dimensional Algebraic Varieties and Vector Bundles, RIMS Kôkyûroku Bessatsu, B9, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 8195.Google Scholar
Fujino, O., Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47(3) (2011), 727789.10.2977/PRIMS/50Google Scholar
Fujino, O., Foundations of the Minimal Model Program, MSJ Memoirs 35, Mathematical Society of Japan, Tokyo, 2017.10.2969/msjmemoirs/035010000Google Scholar
Fujino, O. and Ishitsuka, Y., On the ACC for lengths of extremal rays, Tohoku Math. J. (2) 65(1) (2013), 93103.10.2748/tmj/1365452627Google Scholar
Fujino, O. and Payne, S., Smooth complete toric threefolds with no nontrivial nef line bundles, Proc. Japan Acad. Ser. A Math. Sci. 81(10) (2005), 174179.10.3792/pjaa.81.174Google Scholar
Fujino, O. and Sato, H., Introduction to the toric Mori theory, Michigan Math. J. 52(3) (2004), 649665.Google Scholar
Fujino, O. and Sato, H., Smooth projective toric varieties whose nontrivial nef line bundles are big, Proc. Japan Acad. Ser. A Math. Sci. 85(7) (2009), 8994.10.3792/pjaa.85.89Google Scholar
Fujino, O. and Sato, H., Toric Fano contractions associated to long extremal rays, preprint, 2018, arXiv:1804.05302 [math.AG].Google Scholar
Fujita, T., “On polarized manifolds whose adjoint bundles are not semipositive”, in Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics 10, North-Holland, Amsterdam, 1987, 167178.10.2969/aspm/01010167Google Scholar
Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry.10.1515/9781400882526Google Scholar
Kawamata, Y., On the length of an extremal rational curve, Invent. Math. 105(3) (1991), 609611.10.1007/BF01232281Google Scholar
Kobayashi, S. and Ochiai, T., Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 3147.10.1215/kjm/1250523432Google Scholar
Laterveer, R., Linear systems on toric varieties, Tohoku Math. J. (2) 48 (1996), 451458.10.2748/tmj/1178225343Google Scholar
Lazarsfeld, R., “Positivity in algebraic geometry. I. Classical setting: line bundles and linear series”, in Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics 48, Springer, Berlin, 2004.Google Scholar
Lin, H.-W., Combinatorial method in adjoint linear systems on toric varieties, Michigan Math. J. 51(3) (2003), 491501.10.1307/mmj/1070919555Google Scholar
Matsuki, K., Introduction to the Mori Program, Universitext, Springer, New York, 2002.10.1007/978-1-4757-5602-9Google Scholar
Mustaţǎ, M., Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), 451470.10.2748/tmj/1113247605Google Scholar
Nakayama, N., Zariski-decomposition and Abundance, MSJ Memoirs 14, Mathematical Society of Japan, Tokyo, 2004.10.2969/msjmemoirs/014010000Google Scholar
Oda, T., “Convex bodies and algebraic geometry”, in An Introduction to the Theory of Toric Varieties, Results in Mathematics and Related Areas (3) 15, Springer, Berlin, 1988, Translated from the Japanese.Google Scholar
Payne, S., Fujita’s very ampleness conjecture for singular toric varieties, Tohoku Math. J. (2) 58(3) (2006), 447459.10.2748/tmj/1163775140Google Scholar
Reid, M., “Decomposition of toric morphisms”, in Arithmetic and Geometry, Vol. II, Progress in Mathematics 36, Birkhäuser Boston, Boston, MA, 1983, 395418.10.1007/978-1-4757-9286-7_15Google Scholar
Sato, H., Combinatorial descriptions of toric extremal contractions, Nagoya Math. J. 180 (2005), 111120.10.1017/S0027763000009211Google Scholar