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NormalToricVarieties :: makeSimplicial

makeSimplicial -- make a simplicial toric variety

Synopsis

Description

A normal toric variety is simplical if every cone in its fan is simplicial and a cone is simplicial if its minimal generators are linearly independent over . In fact, the following conditions on a normal toric variety X are equivalent:
  • X is simplicial;
  • every Weil divisor on X has a positive integer multiple that is Cartier;
  • X is -Cartier;
  • the Picard group of X has finite index in the class group of X;
  • X has only finite quotient singularities.

Given a normal toric variety, this method makes a simplicial toric variety with the same rays by triangulating the non-simplicial maximal cones. The triangulation is constructed by repeated regular subdivisions using random integral weight vectors.

i1 : X = normalToricVariety(id_(ZZ^3) | - id_(ZZ^3));
i2 : isSimplicial X

o2 = false
i3 : Y = makeSimplicial X;
i4 : isSimplicial Y

o4 = true
i5 : rays Y === rays X

o5 = true
i6 : max Y

o6 = {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {0, 4, 5}, {1, 2, 3}, {1, 3, 5}, {2,
     ------------------------------------------------------------------------
     3, 6}, {2, 4, 6}, {3, 5, 7}, {3, 6, 7}, {4, 5, 6}, {5, 6, 7}}

o6 : List
i7 : max X

o7 = {{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7},
     ------------------------------------------------------------------------
     {4, 5, 6, 7}}

o7 : List
If the initial toric variety is simplicial, then this method simply returns it.
i8 : PP3 = projectiveSpace 3;
i9 : isSimplicial PP3

o9 = true
i10 : Z = makeSimplicial PP3;
i11 : rays Z === rays PP3

o11 = true
i12 : max Z === max PP3

o12 = true

See also

Ways to use makeSimplicial :