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NormalToricVarieties :: smoothFanoToricVariety(ZZ,ZZ)

smoothFanoToricVariety(ZZ,ZZ) -- get a smooth Fano toric variety from database

Synopsis

Description

This function accesses a database of all smooth Fano toric varieties of dimension at most $6$. The enumeration of the toric varieties follows Victor V. Batyrev's classification ( "On the classification of toric Fano", Journal of Mathematical Sciences (New York), 94 (1999) 1021-1050, arXiv:math/9801107v2 and Hiroshi Sato's "Toward the classification of higher-dimensional toric Fano varieties", The Tohoku Mathematical Journal. Second Series, 52 (2000) 383-413, arXiv:math/9011022) for dimension at most $4$ and Mikkel Ă˜bro's classification ( "An algorithm for the classification of smooth Fano polytopes" arXiv:math/0704.0049v1) for dimensions $5$ and $6$.

There is a unique smooth Fano toric curve, five smooth Fano toric surfaces, eighteen smooth Fano toric threefolds, $124$ smooth Fano toric fourfolds, $866$ smooth Fano toric fivefolds, and $7622$ smooth Fano toric sixfolds.

For all $d$, smoothFanoToricVariety (d,0) yields projective $d$-space.

i1 : PP1 = smoothFanoToricVariety (1,0);
i2 : assert (rays PP1 === rays toricProjectiveSpace 1)
i3 : assert (max PP1 === max toricProjectiveSpace 1)
i4 : PP4 = smoothFanoToricVariety (4,0, CoefficientRing => ZZ/32003, Variable => y);
i5 : assert (rays PP4 === rays toricProjectiveSpace 4)
i6 : assert (max PP4 === max toricProjectiveSpace 4)

The following example was missing from Batyrev's table.

i7 : W = smoothFanoToricVariety (4,123);
i8 : rays W

o8 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {-1, -1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1},
     ------------------------------------------------------------------------
     {0, 0, -1, -1}, {1, 0, 1, 0}, {0, 1, 0, 1}, {-1, -1, -1, -1}}

o8 : List
i9 : max W

o9 = {{0, 1, 5, 6}, {0, 1, 5, 7}, {0, 1, 6, 7}, {0, 2, 4, 6}, {0, 2, 4, 8},
     ------------------------------------------------------------------------
     {0, 2, 6, 8}, {0, 4, 5, 7}, {0, 4, 5, 8}, {0, 4, 6, 7}, {0, 5, 6, 8},
     ------------------------------------------------------------------------
     {1, 2, 3, 7}, {1, 2, 3, 8}, {1, 2, 7, 8}, {1, 3, 5, 6}, {1, 3, 5, 8},
     ------------------------------------------------------------------------
     {1, 3, 6, 7}, {1, 5, 7, 8}, {2, 3, 4, 6}, {2, 3, 4, 7}, {2, 3, 6, 8},
     ------------------------------------------------------------------------
     {2, 4, 7, 8}, {3, 4, 6, 7}, {3, 5, 6, 8}, {4, 5, 7, 8}}

o9 : List

Acknowledgements

We thank Gavin Brown and Alexander Kasprzyk for their help extracting the data for the smooth Fano toric five and sixfolds from their Graded Rings Database.

See also