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NormalToricVarieties :: smoothFanoToricVariety

smoothFanoToricVariety -- get a smooth Fano toric varieties from database

Synopsis

Description

This function accesses a database of all smooth Fano toric varieties of dimension at most 4. The enumeration of the toric varieties follows Victor V. Batyrev's classification; see arXiv:math/9801107v2 and arXiv:math/9011022. There is a unique smooth Fano toric curve, five smooth Fano toric surfaces, eighteen smooth Fano toric threefolds, and 124 smooth Fano toric fourfolds.

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

See also

Ways to use smoothFanoToricVariety :