## Nagoya Mathematical Journal

### Combinatorial descriptions of toric extremal contractions

Hiroshi Sato

#### Abstract

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.

#### Article information

Source
Nagoya Math. J., Volume 180 (2005), 111-120.

Dates
First available in Project Euclid: 14 December 2005

https://projecteuclid.org/euclid.nmj/1134569898

Mathematical Reviews number (MathSciNet)
MR2186671

Zentralblatt MATH identifier
1094.14037

#### Citation

Sato, Hiroshi. Combinatorial descriptions of toric extremal contractions. Nagoya Math. J. 180 (2005), 111--120. https://projecteuclid.org/euclid.nmj/1134569898

#### References

• V. Batyrev, On the classification of smooth projective toric varieties , Tohoku Math. J., 43 (1991), 569–585.
• C. Casagrande, Contractible classes in toric varieties , Math. Z., 243 (2003), 99–126.
• O. Fujino, Notes on toric varieties from Mori theoretic viewpoint , Tohoku Math. J., 55 (2003), 551–564.
• O. Fujino, Equivariant completions of toric contraction morphisms , preprint, math.AG/0311068, to appear in Tohoku Math. J.
• O. Fujino and H. Sato, Introduction to the toric Mori theory , Mich. Math. J., 52 (2004), 649–665.
• W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ (1993).
• A. Kasprzyk, Toric Fano $3$-folds with terminal singularities , preprint, math.AG/ 0311284, to appear in Tohoku Math. J.
• K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York (2002).
• M. Mustaţǎ, Vanishing theorems on toric varieties , Tohoku Math. J., 54 (2002), 451–470.
• T. 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)] 15, Springer-Verlag, Berlin (1988).
• M. Reid, Decomposition of toric morphisms , Arithmetic and geometry, Vol. II, Progr. Math., 36, Birkhäuser Boston, MA (1983), 395–418.
• H. Sato, Toward the classification of higher-dimensional toric Fano varieties , Tohoku Math. J., 52 (2000), 383–413.