Tohoku Mathematical Journal

Toric Fano three-folds with terminal singularities

Alexander M. Kasprzyk

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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.

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Tohoku Math. J. (2), Volume 58, Number 1 (2006), 101-121.

First available in Project Euclid: 18 April 2006

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Zentralblatt MATH identifier

Primary: 14J45: Fano varieties
Secondary: 14J30: $3$-folds [See also 32Q25] 14M25: Toric varieties, Newton polyhedra [See also 52B20] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Toric Fano $3$-folds terminal singularities convex polytopes


Kasprzyk, Alexander M. Toric Fano three-folds with terminal singularities. Tohoku Math. J. (2) 58 (2006), no. 1, 101--121. doi:10.2748/tmj/1145390208.

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