### Toric Fano three-folds with terminal singularities

Alexander M. Kasprzyk
Source: Tohoku Math. J. (2) Volume 58, Number 1 (2006), 101-121.

#### Abstract

This paper classifies all toric Fano $3$-folds with terminal singularities. This is achieved by solving the equivalent combinatorial problem; that of finding, up to the action of $GL(3,\Z)$, all convex polytopes in $\Z^3$ which contain the origin as the only non-vertex lattice point. We obtain, up to isomorphism, $233$ toric Fano $3$-folds possessing at worst $\Q$-factorial singularities (of which $18$ are known to be smooth) and $401$ toric Fano $3$-folds with terminal singularities that are not $\Q$-factorial.

First Page:
Primary Subjects: 14J45
Secondary Subjects: 14J30, 14M25, 52B20
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.tmj/1145390208
Digital Object Identifier: doi:10.2748/tmj/1145390208
Mathematical Reviews number (MathSciNet): MR2221794
Zentralblatt MATH identifier: 1118.14047

### References

A. A. Borisov and L. A. Borisov, Three-dimensional toric Fano varieties with terminal singularities, Text in Russian, Unpublished.
A. A. Borisov and L. A. Borisov, Singular toric Fano three-folds, Russian Acad. Sci. Sb. Math. 75 (1993), 277--283.
Mathematical Reviews (MathSciNet): MR1166957
A. A. Borisov, Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint, arXiv.math.AG/0001109.
D. A. Cox, Update on toric geometry, Geometry of toric varieties, 1--41, Sémin. Congr. 6, Soc. Math. France, Paris, 2002.
Mathematical Reviews (MathSciNet): MR2075606
Zentralblatt MATH: 1050.14001
D. I. Dais, Resolving $3$-dimensional toric singularities, Geometry of toric varietics, 155--186, Sémin. Congr. 6, Soc. Math. France, Paris, 2002.
Mathematical Reviews (MathSciNet): MR2075609
Zentralblatt MATH: 1047.14038
V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97--154.
Mathematical Reviews (MathSciNet): MR495499
G. Ewald, Combinatorial convexity and algebraic geometry, Grad. Texts in Math. 168, Springer, New York, 1996.
Mathematical Reviews (MathSciNet): MR1418400
Zentralblatt MATH: 0869.52001
O. Fujino and H. Sato, Introduction to the toric Mori theory, arXiv.math.AG/0307180v1.
Mathematical Reviews (MathSciNet): MR2097403
Digital Object Identifier: doi:10.1307/mmj/1100623418
Project Euclid: euclid.mmj/1100623418
W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, N.J., 1993.
Mathematical Reviews (MathSciNet): MR1234037
Zentralblatt MATH: 0813.14039
S. Mori, D. R. Morrison and I. Morrison, On four-dimensional terminal quotient singularities, Math. Comp. 51 (1988), 769--786.
Mathematical Reviews (MathSciNet): MR958643
Digital Object Identifier: doi:10.2307/2008778
Zentralblatt MATH: 0712.14026
S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), 133--176.
Mathematical Reviews (MathSciNet): MR662120
Digital Object Identifier: doi:10.2307/2007050
D. R. Morrison, Canonical quotient singularities in dimension three, Proc. Amer. Math. Soc. 93 (1985), 393--396.
Mathematical Reviews (MathSciNet): MR773987
Digital Object Identifier: doi:10.2307/2045598
Zentralblatt MATH: 0533.14001
D. R. Morrison and G. Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984), 15--20.
Mathematical Reviews (MathSciNet): MR722406
Digital Object Identifier: doi:10.2307/2044659
Zentralblatt MATH: 0536.14003
T. Oda, Torus embeddings and applications, Tata Inst. Fund. Res. Lectures on Math. and Phys. 57, Tata Inst. Fund. Res., Bombay by Springer, Berlin-New York, 1978.
Mathematical Reviews (MathSciNet): MR546291
Zentralblatt MATH: 0417.14043
M. Reid, Minimal models of canonical $3$-folds, Algebraic varieties and analytic varieties, (Tokyo, 1981), 131--180, Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983.
Mathematical Reviews (MathSciNet): MR715649
Zentralblatt MATH: 0558.14028
M. Reid, Young person's guide to canonical singularities, Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345--414, Proc. Sympos. Pure Math. 46, pt. 1, Amer. Math. Soc., Providence, R.I., 1987.
Mathematical Reviews (MathSciNet): MR927963
Zentralblatt MATH: 0634.14003
H. E. Scarf, Integral polyhedra in three space, Math. Oper. Res. 10 (1985), 403--438.
Mathematical Reviews (MathSciNet): MR798388
K. Suzuki, On Fano indicies of $\mathbbQ$-fano $3$-folds, arXiv.math.AG/0210309v1.
J. A. Wiśniewski, Toric Mori theory and fano manifolds, Geometry of toric varieties, 249--272, Sémin. Congr. 6, Soc. Math. France, Paris, 2002.
Mathematical Reviews (MathSciNet): MR2063740