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NormalToricVarieties :: normalToricVariety(Fan)

normalToricVariety(Fan) -- make a normal toric variety from a 'Polyhedra' fan

Synopsis

Description

This method makes a NormalToricVariety from a Fan as implemented in the Polyhedra package.

i1 : F = faceFan convexHull (id_(ZZ^3) | -id_(ZZ^3))

o1 = F

o1 : Fan
i2 : rays F

o2 = | -1 1 0  0 0  0 |
     | 0  0 -1 1 0  0 |
     | 0  0 0  0 -1 1 |

              3        6
o2 : Matrix ZZ  <--- ZZ
i3 : maxCones F

o3 = {{0, 2, 4}, {1, 2, 4}, {0, 3, 4}, {1, 3, 4}, {0, 2, 5}, {1, 2, 5}, {0,
     ------------------------------------------------------------------------
     3, 5}, {1, 3, 5}}

o3 : List
i4 : X = normalToricVariety F;
i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F)

The recommended method for creating a NormalToricVariety from a fan is normalToricVariety(List,List). In fact, this package avoids using objects from the Polyhedra package whenever possible. Here is a trivial example, namely projective 2-space, illustrating the substantial increase in time resulting from the use of a Polyhedra fan.

i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
     -- used 0.000014491 seconds

o6 = normalToricVariety((({{-1, -1}, {1, 0}, {0, 1}}(,({{0, 1}, {0, 2}, {1, 2}} )))))

o6 : NormalToricVariety
i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
     -- used 0.0247353 seconds
i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)

See also