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Combinatorial descriptions of toric extremal Contractions

Published online by Cambridge University Press:  11 January 2016

Hiroshi Sato
Hiroshi Sato*
Affiliation:
Department of Mathematics Tokyo Institute of Technology 2-12-1 Oh-Okayama Meguro-ku Tokyo, 152-8551Japanhirosato@math.titech.ac.jp
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Abstract

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In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties are not complete. It is well-known that the complete case is settled by using Reid’s wall theory which can not be applied to the non-complete case. Therefore, we can achieve them by using the notion of extremal primitive relations. As applications, we can generalize some of Mustaţă’s results related to Fujita’s conjecture on toric varieties for the relative case.

Keywords

14M2514E30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 180 , 2005 , pp. 112 - 120
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[B] Batyrev, V., On the classification of smooth projective toric varieties, Tohoku Math. J., 43 (1991), 569–585.Google Scholar
[C] Casagrande, C., Contractible classes in toric varieties, Math. Z., 243 (2003), 99–126.CrossRefGoogle Scholar
[Fj1] Fujino, O., Notes on toric varieties from Mori theoretic viewpoint, Tohoku Math. J., 55 (2003), 551–564.Google Scholar
[Fj2] Fujino, O., Equivariant completions of toric contraction morphisms, preprint, math.AG/031 1068, to appear in Tohoku Math. J. Google Scholar
[FS] Fujino, O. and Sato, H., Introduction to the toric Mori theory, Mich. Math. J., 52 (2004), 649–665.Google Scholar
[F1] Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.Google Scholar
[K] Kasprzyk, A., Toric Fano 3-folds with terminal singularities, preprint, math.AG/ 0311284, to appear in Tohoku Math. J. Google Scholar
[Ma] Matsuki, K., Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002.Google Scholar
[Mu] Mustaţă, M., Vanishing theorems on toric varieties, Tohoku Math. J., 54 (2002), 451–470.Google Scholar
[O] Oda, T., 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)] 15, Springer-Verlag, Berlin, 1988.Google Scholar
[R] Reid, M., Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., 36, Birkhäuser Boston, MA (1983), pp. 395–418.Google Scholar
[S] Sato, H., Toward the classification of higher-dimensional toric Fano varieties, To-hoku Math. J., 52 (2000), 383–413.Google Scholar