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NormalToricVarieties :: smoothFanoToricVariety

smoothFanoToricVariety -- 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 : rays PP1      

o2 = {{-1}, {1}}

o2 : List
i3 : max PP1

o3 = {{0}, {1}}

o3 : List
i4 : PP4 = smoothFanoToricVariety(4,0);
i5 : rays PP4

o5 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0,
     ------------------------------------------------------------------------
     1}}

o5 : List
i6 : max PP4

o6 = {{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}}

o6 : List
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

Ways to use smoothFanoToricVariety :