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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 (see arXiv:math/9801107v2 and arXiv:math/9011022) for dimension at most 4 and Mikkel Ă˜bro’s classification (see 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